# Near Classification of Compact Hyperbolic Coxeter $d$-Polytopes with $d+4$
Facets and Related Dimension Bounds

Amanda Burcroff  Department of Mathematics, Harvard University
<EMAIL_ADDRESS>
###### Abstract.

We complete the classification of compact hyperbolic Coxeter $d$-polytopes
with $d+4$ facets for $d=4$ and $5$. By previous work of Felikson and
Tumarkin, the only remaining dimension where new polytopes may arise is $d=6$.
We derive a new method for generating the combinatorial type of these
polytopes via the classification of point set order types. In dimensions $4$
and $5$, there are $348$ and $51$ polytopes, respectively, yielding many new
examples for further study.

We furthermore provide new upper bounds on the dimension $d$ of compact
hyperbolic Coxeter polytopes with $d+k$ facets for $k\leq 10$. It was shown by
Vinberg in $1985$ that for any $k$, we have $d\leq 29$, and no better bounds
have previously been published for $k\geq 5$. As a consequence of our bounds,
we prove that a compact hyperbolic Coxeter $29$-polytope has at least $40$
facets.

##  1\. Introduction

Let $\mathbb{H}^{d}$ be the $d$-dimensional real hyperbolic space. A
hyperbolic Coxeter polytope is a domain in $\mathbb{H}^{d}$ bounded by a
collection of geodesic hyperplanes, such that each intersecting pair of
hyperplanes meets at dihedral angle $\frac{\pi}{m}$ for some integer $m\geq
2$. Hyperbolic Coxeter polytopes are precisely the fundamental domains of
discrete hyperbolic reflection groups. These polytopes also have relevance to
the construction of orbifolds and manifolds, in particular some of minimal
volume [21].

In this paper, we classify the compact hyperbolic Coxeter $d$-polytopes with
$d+4$ facets for $d=4$ and $5$, as well as improve some bounds on the
dimension of compact Coxeter polytopes with few facets. While Euclidean and
spherical Coxeter polytopes were classified by Coxeter in 1934 [9], no
complete classification is known in the hyperbolic case. Henceforth, all
polytopes are assumed to be hyperbolic unless otherwise specified.

The partial classification of compact Coxeter polytopes has been obtained by
restricting either the dimension, combinatorial type, or number of facets. A
dynamic summary of this progress is maintained by Anna Felikson on her webpage
www.maths.dur.ac.uk/users/anna.felikson/Polytopes/polytopes.html.

This work is based on the author’s master’s thesis [8] at Durham University,
supervised by Pavel Tumarkin. The thesis is publicly available at
http://etheses.dur.ac.uk/14202/.

###  1.1. Classification of compact Coxeter $d$-polytopes with $d+4$ facets
for $d\neq 6$

We first focus on restricting the number of facets with respect to the
dimension. Compact Coxeter simplices, i.e., $d$-polytopes with $d+1$ facets,
were classified by Lannér in 1950 [22]. These arise only in dimensions $2$,
$3$, and $4$. The compact Coxeter $d$-polytopes with $d+2$ facets are
classified in [20] and [11]; these arise in dimensions $3$ through $5$ and
must be a simplicial prism except in dimension $4$. Esselmann [10] showed in
1994 that a compact Coxeter $d$-polytope with $d+3$ facets must satisfy $d\leq
8$, and that there is a unique polytope of dimension $8$ (first constructed by
Bugaenko [7]). In 2007, Tumarkin [28] completed the classification of compact
Coxeter $d$-polytopes with $d+3$ facets, which arise in dimensions $2$ through
$6$ and $8$.

The first portion of this paper is dedicated to furthering the classification
of compact Coxeter $d$-polytopes with $d+4$ facets. In 2008, Felikson and
Tumarkin [12] showed that such polytopes arise only in dimension at most $7$,
and furthermore that there is a unique compact Coxeter $7$-polytope with $11$
facets (originally constructed by Bugaenko [7]). We complete this
classification in dimensions $4$ and $5$, with $348$ polytopes in dimension
$4$ and $51$ of dimension $5$. This includes the first known Coxeter polytope
in dimension $>3$ with an angle of less than $\frac{\pi}{10}$ and the first
known Coxeter polytope in dimension $>3$ with an angle of $\frac{\pi}{7}$,
along with many new essential polytopes. A polytope is _essential_ if it is
minimal with respect to the operations of taking the fundamental domain of a
finite index reflection subgroup of the corresponding reflection group, or
gluing two Coxeter polytopes along congruent facets (see [15] for further
details). The present work combined with that of Felikson and Tumarkin yields
a classification in all dimensions except $6$, where the only known polytope
was constructed by Bugaenko [7]. We show that a compact Coxeter $6$-polytope
with $10$ facets must contain a missing face of size $3$ or $4$.

In order to obtain this classification, we develop a new method for
restricting the possible combinatorial types of these polytopes. The Gale
diagram of a $d$-dimensional polytope with $n$ facets is an $n-d-1$
arrangement of points in Euclidean space that encodes the combinatorial type
of the polytope. In studying $d$-polytopes with $d+4$ facets, this means their
combinatorial types can be studied in terms of a $3$-dimensional point
arrangement. Moreover, each Gale diagram can be transformed into an affine
Gale diagram, an arrangement of positive and negative points, to further
reduce the dimension by $1$ [29]. We show that the (affine) Gale diagrams of
compact $d$-dimensional hyperbolic polytopes with $d+4$ facets can be
generated by bipartitioning the points in all point set order types with $d+4$
points (see Theorem 3.5 for further details). The point set order types of
sizes up to $10$ have been enumerated and made available through the Point Set
Order Type Database [2]. Using this database, we produce a reasonably short
list of possible combinatorial types for the polytopes of interest in
dimensions $4$ and $5$. In dimension $6$, the same methods can be applied, but
the number of point set order types makes the process rather computationally
demanding.

Having greatly restricted the possible combinatorial types in dimensions $4$
and $5$, we then determine whether each combinatorial type can be realised as
one or more polytopes. This involves enumerating weighted graphs with
restrictions on certain subgraphs and the spectral properties of their
adjacency matrices. Though the search space is infinite, combinatorial and
linear algebraic techniques (see, e.g., [29]) have previously been successful
in reducing this to a computational problem. In particular, Tumarkin handled
the analogous task for polytopes with $d+3$ facets by inspecting local
determinants, face structures, and gluings of Lannér diagrams [28]. These
methods are only partially effective for the polytopes with $d+4$ facets, due
to the greater complexity of the polytopes and less restrictive missing face
structures. We then utilise the computer algebra system Mathematica to check a
finite number of cases in order to list all polytopes of a given combinatorial
type, a technique which was recently used in classifying the compact Coxeter
cubes [19].

Ma and Zheng independently and via different methods classified the compact
hyperbolic Coxeter $4$-polytopes with $8$ facets [23] and $5$-polytopes with
$9$ facts [24]. Their work became publicly available within a few months
following the release of the author’s master’s thesis on this topic (available
at http://etheses.dur.ac.uk/14202/). The author is very grateful to Ma and
Zheng for their communication about this classification, as it helped to
correct several minor errors. Due to the sheer volume of data handling
required by both our methods and those of Ma-Zheng, there were a few errors in
the polytope lists initially announced by both groups. In the case of the
author’s master’s thesis, coding errors led to the omission of ten
$4$-polytopes of type $G_{2}$ (all obtainable from one such polytope by gluing
prisms) and one $5$-polytope of type $H_{6}$. There were also three polytopes
of type $G_{3}$ that were double-counted due to isomorphic Coxeter diagrams
being listed. Ma, Zheng, and the current author now agree on the published
lists. The existence of two independent methods for obtaining these polytopes
may lend some confidence to the accuracy of this rather delicate
classification.

###  1.2. Bounding the dimension of polytopes with few facets

In Sections 9 and 10, we shift our focus to bounding the dimension of certain
compact Coxeter polytopes. It was shown in 1984 by Vinberg [30] that compact
Coxeter polytopes do not arise in dimensions higher than $29$. Vinberg
proceeded by constructing certain weightings on the edges of the polytopes,
and utilised a result of Nikulin [26] on the average number of vertices along
a $2$-dimensional face. In Section 9, we show that a slight modification of
Vinberg’s argument yields a stronger bound for $3$-free polytopes, that is,
polytopes having missing faces only of order $2$. In particular, we show that
compact Coxeter $3$-free polytopes do not arise in dimension higher than $13$.
111Since the release of this work, our upper bound of $13$ has been improved
using similar methods (which, in turn, were inspired by those of Vinberg [30])
to $12$ by Alexandrov [3].

In Section 10, we improve the bounds on the dimension of compact Coxeter
$d$-polytopes with $d+k$ facets for $5\leq k\leq 10$. In order to obtain an
initial bound, we examine certain faces which must themselves be compact
Coxeter polytopes, similar to the methods used by Felikson and Tumarkin [12]
to bound the dimension when $k=4$. These ideas combined with the results in
Section 9 yield the bounds in Theorem 10.7. The rest of the section is devoted
to improving these bounds in certain cases, frequently referring to the
classification of polytopes with fewer facets. One corollary of our improved
bounds is that any compact Coxeter polytopes of dimension $29$, i.e., the
threshold of Vinberg’s bound, must have at least $29+11=40$ facets.

##  2\. Convex Polytopes and Their Combinatorial Types

###  2.1. Convex polytope preliminaries

A convex polytope in $\mathbb{H}^{d}$ can be defined as $\bigcap_{i\in
I}H_{i}^{-}$, where $I$ is an index set and $H_{i}^{-}$ is a half-space
containing $P$. Throughout this paper we will consider only polytopes of
finite volume, or equivalently, those which can be obtained as the convex hull
of a finite point set. We furthermore assume that the set of bounding
hyperplanes is chosen minimally to define $P$. The intersection of a convex
polytope with a bounding hyperplane is called a _facet_ of the polytope. We
identify the index set $I$ with the set $\\{0,1,\dots,d+k-1\\}$, where $d+k$
is the number of facets of $P$. If the convex polytope $P$ has a vertex at
infinity, then $P$ is said to be _non-compact_ , otherwise we say $P$ is
_compact_. For the purposes of this section, we do not assume $P$ is compact,
though we only consider compact polytopes in the remainder of this paper. If
each vertex is formed by the intersection of precisely $d$ half spaces, then
$P$ is said to be _simple_. An equivalent condition is that every $(d-j)$-face
is contained in precisely $j$ facets.

Fix an enumeration of the facets of $P$ as $f_{0}$, $f_{1}$, …, $f_{d+k-1}$.
Then each face $P\cap f_{i_{0}}\cap\cdots\cap f_{i_{s}}$ for $0\leq
i_{0}<\dots<i_{s}<d+k$ is denoted by the string $i_{0}\dots i_{s}$. A _missing
face_ of $P$ is a list of facets whose intersection is empty, but such that
the intersection of every proper subset of these facets is non-empty. That is,
a missing face is an intersection of facets $i_{0}\dots i_{s}$ that is empty,
but such that $i_{0}\dots i_{m-1}i_{m+1}\dots i_{s}$ is non-empty for each
$0\leq m\leq s$. We refer to the set of missing faces of $P$ as the _missing
face list_. Note that every missing face list is an antichain by inclusion,
i.e., no missing face is a subset of another. Two polytopes are said to be
_isomorphic_ , or of the same _combinatorial type_ , provided there exists a
bijective correspondence between their faces, such that two faces of the first
polytope meet if and only if the corresponding faces of the second meet. In
particular, two polytopes have the same combinatorial type if and only if they
have the same set of missing faces, up to relabelling of the facets.

###  2.2. Gale diagrams and affine Gale diagrams

An important technique in classifying convex polytopes is representing a
polytope by a “diagram” from which one can read off the face structure.
Throughout this paper, we frequently reference _Gale diagrams_ , introduced in
a $1956$ paper by David Gale [16]. Note that while Gale diagrams are often
defined using the vertices of a polytope, we look at the dual construction
defined on the facets. We will consider only simple polytopes in this section,
which include all compact Coxeter polytopes (see Remark 4.5).

Given a simple $d$-polytope $P$ with $d+k$ facets, a Gale diagram of $P$
consists of a set of $d+k$ points on $S^{k-2}\subset\mathbb{R}^{k-1}$
corresponding to the facets of $P$. These points can be obtained by applying a
_Gale transform_ to the affine dependences of the normal vectors to the facets
(see [17] for details of the construction). These points encode the face
structure of $P$ in the following way: a set of facets $\\{f_{i}:i\in I\\}$ of
$P$ has a non-trivial intersection if and only if the points in the Gale
diagram corresponding to $\\{f_{j}:j\in[d+k]\operatorname{\setminus}I\\}$ have
a convex hull containing the origin. Two Gale diagrams are _isomorphic_ if
their corresponding polytopes are combinatorially equivalent.

A set of $d+k$ points on $S^{k-2}$ corresponds to a Gale diagram of a simple
convex $d$-polytope with $d+k$ facets if and only if every half space bounded
by a hyperplane through the origin contains at least two of the points. One
can see that the latter condition is necessary by taking the convex hull of
the points corresponding to $\\{f_{j}:j\neq i\\}$ for any $i\in[d+k]$; this
convex hull should contain the origin, as any facet is itself a non-empty
face.

In the case $k=4$, Gale diagrams consist of point configurations on
$S^{2}\subseteq\mathbb{R}^{3}$. These seem rather difficult to classify, and
thus a crucial step in our analysis is passing to the affine Gale diagrams.
These affine variants encode the same information as Gale diagrams, but using
partitioned sets of points in $\mathbb{R}^{k-2}$. Thus, the configurations are
reduced to points on the Euclidean plane when $k=4$, which can be classified
using point set order types (further details in Section 3).

An _affine Gale diagram_ of a $d$-polytope with $d+k$ facets consists of two
(not necessarily disjoint) point sets, called “positive” and “negative”, in
$\mathbb{R}^{k-2}$ containing $d+k$ points in total (counted with
multiplicity). An affine Gale diagram is obtained from a Gale diagram $G$ by
taking a hyperplane $H$ through the origin not containing any points of $G$,
and projecting the points orthogonally onto $H$, with the projections of
points from the open half space $H^{+}$ being labelled “positive” and those
from $H^{-}$ labelled “negative”. The face structure of $P$ is determined in
the following way: a set of facets $\\{f_{i}:i\in I\\}$ of $P$ has a non-
trivial intersection if and only if the convex hull of the positive points in
$\\{f_{j}:j\notin I\\}$ non-trivially intersects the convex hull of the
negative points in $\\{f_{j}:j\notin I\\}$.

A configuration of positive and negative points then corresponds to an affine
Gale diagram of a polytope if and only if, for every hyperplane through the
origin $H$ in $\mathbb{R}^{k-2}$, the total number of positive points
contained in the open half space $H^{+}$ and negative points contained in
$H^{-}$ is at least $2$. This classification follows directly from the
analogous half space condition for Gale diagrams.

##  3\. Combinatorial Types of Simple $d$-Polytopes with $d+4$ Facets

###  3.1. Affine Gale diagrams of simple $d$-polytopes with $d+4$ facets

In order to determine the compact hyperbolic $d$-polytopes with $d+4$ facets
in dimension $4$ and $5$, our methods involve first limiting their possible
combinatorial types. This was accomplished for compact hyperbolic
$d$-polytopes with $d+3$ facets by Tumarkin [28], using _standard Gale
diagrams_ described by Esselmann[10]. However, Tumarkin’s methods do not seem
immediately generalisable to Gale diagrams of higher dimension. We develop a
new method for generating affine Gale diagrams using a classification of point
set order types by Aichholzer, Aurenhammer, and Krasser [1]. These are reduced
to a representative list of Gale diagrams, from which one can easily determine
the set of missing faces. Later sections are devoted to determining which of
these combinatorial types are realisable.

We now describe how to obtain the potential combinatorial types of compact
hyperbolic $d$-polytopes with $d+4$ facets, listed in Appendix A. First, we
show that every combinatorial type of such a polytope has a corresponding
affine Gale diagram satisfying certain properties (see Subsection 2.2 for
details on affine Gale diagrams). In fact, the conditions we consider hold for
the more general case of simple hyperbolic $d$-polytopes with $d+4$ facets.

######  Remark 3.1.

Let $P$ be a simple $d$-polytope with $d+k$ facets for $k\geq 2$. Observe that
the positive and negative points of an affine Gale diagram for $P$ are
necessarily disjoint. If not, let $v\in\mathbb{R}^{2}$ be a point which is
both positive and negative. The facets not corresponding to this point then
intersect non-trivially in a face of codimension $d+k-2\geq d$, which is not
possible.

A set of points in $\mathbb{R}^{d}$ is said to be in _general position_ if no
$m$ points lie in a subspace of dimension $m-2$ for $m=2,\dots,d+1$. In
particular, when $d=2$ this is equivalent to requiring that no three points
are collinear.

######  Proposition 3.2.

For $d\geq 4$, every simple $d$-polytope with $d+4$ facets admits an affine
Gale diagram where all points are in general position.

###### Proof.

Let $A=A_{+}\cup A_{-}$ be an affine Gale diagram for $P$, where
$A_{+},A_{-}\subseteq\mathbb{R}^{2}$ denote the set of positive points and the
set of negative points, respectively. Our aim will be to slightly perturb the
points of $A$ to ensure they are in general position while preserving the
associated combinatorial type. By Remark 3.1, $A_{+}$ and $A_{-}$ are
necessarily disjoint.

Since every vertex of $P$ is obtained as the intersection of exactly $d$
facets, then no set of fewer than $4$ positive and negative vertices can have
intersecting convex hulls. In particular, no positive vertex lies on the line
segment between two negative vertices, and no negative vertex lies on the line
segment between two positive vertices. Thus, the convex hulls of a set of
positive and negative vertices intersect non-trivially if and only if their
interiors intersect.

Thus, each point of $A$ can be moved within a small neighbourhood of its
original position without changing whether a given set of positive and
negative points have intersecting convex hulls, i.e., without changing the
combinatorial type associated to $A$. We can thus slightly perturb the points
within this small neighbourhood to obtain a set of points in general position.
This process will yield an affine diagram $A^{\prime}=A^{\prime}_{+}\cup
A^{\prime}_{-}$ of the same combinatorial type as $A$ but with all points in
general position, along with the additional property that
$|A^{\prime}_{+}|=|A_{+}|$. ∎

It has been shown by Tumarkin and Felikson [13, 14] that for $k\geq 4$, every
compact $d$-polytope with $d+k$ facets has at least two pairs of non-
intersecting facets (see Theorem 5.1). Given a polytope $P$ with $d+4$ facets,
let $f$ and $f^{\prime}$ be a non-intersecting pair of facets. In any Gale
diagram associated to $P$, the points corresponding to $f$ and $f^{\prime}$
can be separated from the remaining points by a hyperplane through the origin.
Take the affine Gale diagram obtained by orthogonal projection onto this
hyperplane and choosing the half-space containing $f$ and $f^{\prime}$ to be
positive. Applying the results of Proposition 3.2, in particular the last line
of the proof, we obtain an affine Gale diagram associated to $P$ where all
points are in general position and with exactly two positive points.

######  Proposition 3.3.

Let $A=A_{+}\cup A_{-}$ be an affine Gale diagram for a simple $d$-polytope
with $d+4$ facets, where $A$ is in general position in $\mathbb{R}^{2}$ and
$|A_{+}|=2$. Then $A_{+}$ is contained in the interior of the convex hull of
$A_{-}$.

###### Proof.

Suppose that $u\in A_{+}$ is not in the interior of the convex hull of
$A_{-}$. Then there is some $v\in A_{-}$ such that no two points of $A_{-}$
are separated by the line through $u$ and $v$. In other words, all points of
$A_{-}$ lie in a closed half space $H\cup H^{+}$ bounded by the line $H$
through $u$ and $v$. Observe then that the number of positive points in
$H^{+}$ is at most one, as $|A_{+}|=2$ and $u\in H$. Moreover, there are no
negative points in $H^{-}$. As discussed in Subsection 2.2, $|A_{+}\cap
H^{+}|+|A_{-}\cap H^{-}|$ must be at least $2$ for $A$ to be an affine Gale
diagram, so we reach a contradiction. ∎

In summary, every simple $d$-polytope with $d+4$ facets admits an affine Gale
diagram $A=A_{+}\cup A_{-}$ where

  1. (i)

all points of $A$ are in general position,

  2. (ii)

$|A_{+}|=2$, and

  3. (iii)

$A_{+}$ is contained in the interior of the convex hull of $A_{-}$.

In the sequel, we use these restrictions to show that all affine Gale diagrams
of compact $d$-polytopes with $d+4$ facets can be obtained by bipartitioning
the $d+k$ points of a point set order type.

###  3.2. Point set order types

In order to enumerate the possible Gale diagrams of compact $d$-polytopes with
$d+4$ facets, we utilise a classification of the point set order types of size
at most $10$ [1]. This classification was obtained with the aid of extensive
computer search in 2002 and is restricted to point sets in general position.
The order type of a point set records the relative orientations of triples of
points, from which one can determine many other combinatorial properties, such
as whether a set of points is in convex position or whether line segments
between points intersect.

The _orientation_ of an ordered triple
$((x_{1},x_{2}),(y_{1},y_{2}),(z_{1},z_{2}))\in(\mathbb{R}^{2})^{3}$ of points
in general position is determined by the sign of the determinant of the matrix

$\begin{bmatrix}1&1&1\\\ x_{1}&y_{1}&z_{1}\\\
x_{2}&y_{2}&z_{2}\end{bmatrix}\,.$

The points are said to be oriented _counter-clockwise_ if the determinant is
positive and _clockwise_ if the determinant is negative. The _order type_ of a
set of points $\\{p_{1},\dots,p_{n}\\}\subset\mathbb{R}^{2}$ in general
position is a mapping that assigns to each ordered triple $(i,j,k)$ of
distinct elements in $\\{1,\dots,n\\}$ the orientation of the point triple
$(p_{i},p_{j},p_{k})$. One can also define the order type of points not in
general position, but we are not concerned with those in the present setting.
Two finite subsets $P,Q\subset\mathbb{R}^{2}$ are said to be _combinatorially
equivalent_ if they have the same order type, i.e., if there exists a
bijection $f:P\to Q$ that preserves orientations.

The following lemma shows that the order type of a point set records the same
intersection properties of convex hulls that the affine Gale diagrams use.
This is a crucial step in seeing that all affine Gale diagrams of compact
$d$-polytopes with $d+4$ facets can be obtained by partitioning the points of
each order type.

######  Lemma 3.4.

Fix two combinatorially equivalent point sets $P,Q\subseteq\mathbb{R}^{2}$,
each in general position, with an orientation-preserving bijection
$\sigma:P\to Q$ between them. Then for any $P_{1},P_{2}\subseteq P$ and
$Q_{1}=\sigma(P_{1}),Q_{2}=\sigma(P_{2})\subseteq Q$, we have

$\operatorname{conv}(P_{1})\cap\operatorname{conv}(P_{2})=\emptyset\iff\operatorname{conv}(Q_{1})\cap\operatorname{conv}(Q_{2})=\emptyset\,.$

###### Proof.

Note that we can assume $P_{1}$ and $P_{2}$, hence $Q_{1}$ and $Q_{2}$, are
disjoint, since otherwise both intersections are clearly non-empty. Since the
sets are in general position, $\operatorname{conv}(P_{1})$ and
$\operatorname{conv}(P_{2})$ intersect non-trivially if and only if their
interiors intersect. If their interiors intersect, there are two
possibilities, up to swapping $P_{1}$ and $P_{2}$:

  1. (i)

there exist points $u,v\in P_{1}$ and $x,y\in P_{2}$ such that the line
segment $\overline{uv}$ transversally intersects the line segment
$\overline{xy}$, i.e, the segments intersect at an interior point, or

  2. (ii)

$\operatorname{conv}(P_{1})$ is contained in the interior of
$\operatorname{conv}(P_{2})$.

We can detect the first condition by looking at the orientations of certain
triples. The line segments $\overline{uv}$ and $\overline{xy}$ intersect
transversally if and only if $(u,v,x)$ and $(u,v,y)$ have opposite
orientations, and additionally $(x,y,u)$ and $(x,y,v)$ have opposite
orientations.

The second condition can also be detected by the order type. Recall that a set
of points in $\mathbb{R}^{2}$ is in convex position, i.e., form the vertices
of a convex polygon, if and only if every set of four points is in convex
position (this follows directly from Caratheodory’s theorem; see, e.g., [25]).
The latter condition is equivalent to saying that every four points define
precisely one pair of intersecting line segments, which can be detected by the
order type as described in the previous paragraph. Moreover, given a set of
points $(p_{1},\dots,p_{n})$ forming cyclically adjacent vertices of a convex
polygon, hence in convex position, a point $p$ is contained in their convex
hull if and only if the orientations of $(p_{i},p_{i+1},p)$ are all the same,
where the subscripts are taken modulo $n$. Now, if
$\operatorname{conv}(P_{1})$ is contained in $\operatorname{conv}(P_{2})$,
then there is a subset $P_{2}^{\prime}\subset P_{2}$ in convex position such
that $\operatorname{conv}(P_{1})\subseteq\operatorname{conv}(P_{2}^{\prime})$.
Using the order type, we can then detect if each vertex of $P_{1}$ is
contained in the convex hull of $P_{2}^{\prime}$, hence the convex hull of
$P_{2}$.

Therefore, whether $\operatorname{conv}(P_{1})\cap\operatorname{conv}(P_{2})$
is empty is determined by the orientations of triples in $P_{1}\cup P_{2}$.
That is, whether each such intersection is non-empty is determined by the
order type of $P$. ∎

We now obtain our method for generating the desired affine Gale diagrams using
the point set order types.

######  Theorem 3.5.

Every compact Coxeter $d$-polytope $P$ with $d+4$ facets admits an affine Gale
diagram obtained by taking an arrangement of $d+4$ points
$A\subseteq\mathbb{R}^{2}$ in general position and choosing two points from
the interior of $\operatorname{conv}(A)$ to be positive. Moreover, the
combinatorial type of $P$ is completely determined by the order type of $X$.

###### Proof.

This follows directly from Proposition 3.2, Proposition 3.3, and Lemma 3.4. ∎

An example of such an affine Gale diagram and a polytope realising its
combinatorial type are depicted in Figure 1.

Figure 1. On the left is an affine Gale diagram of combinatorial type $G_{6}$,
with the positive points depicted in red and the negative points in black. The
line segments between negative points have been included to aid in identifying
the convex hull of each subset. On the right, we have a Coxeter diagram of a
compact $4$-polytope with combinatorial type $G_{6}$ (see Appendix A for a
description of this type); the labelling of the vertices corresponding to the
$8$ facets is preserved (see Section 4 for details of how to interpret Coxeter
diagrams).

Using the order type database for up to $10$ points [1], one can classify the
combinatorial types of all polytopes with $d+4$ facets in dimension at most
$6$. There are $3{,}315$ order types on $8$ points, which yield $34$ possible
combinatorial types of simple $4$-polytopes with $8$ facets having at least
one pair of disjoint facets, listed in Table A. The classification in
dimension $4$ was also completed by Grünbaum and Sreedharan [18] via different
methods, though they also considered polytopes with no pairs of disjoint
facets. There are $158{,}817$ order types on $9$ points, which yield $186$
possible combinatorial types of $5$-polytopes with $9$ facets; the $111$
combinatorial types with at least two pairs of disjoint facets are listed in
Appendix A. Note that not all of these combinatorial types are realized; this
is merely a short list of possible types from which we determine those that
are realized as compact Coxeter polytopes.

There are $14{,}309{,}547$ order types on $10$ facets; the significant jump in
the number of order types, recalling that we must also iterate over all
choices of two positive vertices for each order type, makes determining the
possible combinatorial types for this class computationally challenging. The
author plans to handle this class in a later work. However, we have been able
to glean some partial information about $6$-polytopes with 10 facets; see
Section 8 for more details.

##  4\. Coxeter Diagrams and Gram Matrices

In this section, we define (abstract) Coxeter diagrams and Gram matrices. We
also illustrate several properties of these structures when associated to a
compact hyperbolic Coxeter $d$-polytope with $d+4$ facets, which are used in
our classification of these polytopes in the subsequent sections.

We begin by introducing the notion of an _abstract Coxeter diagram_ , a
weighted graph which can, under certain conditions, describe the dihedral
angles of a polytope.

######  Definition 4.1.

An _abstract Coxeter diagram_ is a finite simple graph (i.e., one-dimensional
simplicial complex) with weighted edges. The weights $w_{ij}$ must be positive
and satisfy the following condition: if $w_{ij}<1$, then
$w_{ij}=\cos\left(\frac{\pi}{m_{ij}}\right)$ for some integer $m_{ij}\geq 3$.

We often denote abstract Coxeter diagrams by the letter $\Sigma$. The order
$|\Sigma|$ of the diagram $\Sigma$ is the number of vertices of $\Sigma$. A
_subdiagram_ $\Sigma^{\prime}$ of $\Sigma$ is a vertex-induced subgraph of
$\Sigma$, where the edge weights are preserved; this relation is written as
$\Sigma^{\prime}\subset\Sigma$. For any two subdiagrams
$\Sigma_{1},\Sigma_{2}$ of $\Sigma$, we let
$\langle\Sigma_{1},\Sigma_{2}\rangle$ denote the subdiagram induced by the
vertices contained in either $\Sigma_{1}$ or $\Sigma_{2}$. When drawing
abstract Coxeter diagrams, we adhere to the following conventions:

  1. (i)

If $w_{ij}<1$, hence $w_{ij}=\cos\left(\frac{\pi}{m_{ij}}\right)$ for some
integer $m_{i,j}\geq 2$, then the corresponding edge is drawn as a straight
line labelled by $m_{ij}$. In the special cases where $m_{ij}$ is equal to
$3$, $4$, or $5$, we draw the corresponding edge as an unlabelled single,
double, or triple line, respectively. If $m_{ij}=2$, we leave the edge empty.

  2. (ii)

If $w_{ij}>1$, the corresponding edge is drawn as a dashed line with label
$w_{ij}$.

  3. (iii)

If $w_{ij}=1$, the corresponding edge is drawn as an unlabelled bold line.
Since we are considering only compact polytopes, this case does not arise in
our setting.

An ordinary edge of weight $\cos\left(\frac{\pi}{m}\right)$ for some $m\geq 2$
is said to have _multiplicity_ $m-2$. We refer to an ordinary edge of weight
$\cos\left(\frac{\pi}{m_{ij}}\right)$ for $2\leq m_{ij}\leq 5$ as having _low
weight_ , and an ordinary edge of weight $\cos\left(\frac{\pi}{m_{ij}}\right)$
for $m_{ij}\geq 6$, i.e., having multiplicity at least $4$, as being _multi-
multiple_. While abstract Coxeter diagrams encode the information of the
dihedral angles in a graph theoretic context, it is also often useful to treat
these linear-algebraically. We do so by considering a certain weighted
adjacency matrix, known as a Gram matrix.

######  Definition 4.2.

The _Gram matrix_ $M(\Sigma)$ of an abstract Coxeter diagram $\Sigma$ on $n$
vertices is an $n\times n$ matrix with entries $m_{ij}$ as prescribed:

$m_{ij}=\begin{cases}1&\text{ if }i=j\,,\\\ -w_{ij}&\text{ if vertices
}i,j\in\Sigma\text{ are connected by an edge of weight $w_{ij}$}\,,\\\
0&\text{ if vertices }i,j\in\Sigma\text{ are not adjacent}\,.\end{cases}$

Now that we have introduced the notions of Coxeter diagrams and Gram matrices
abstractly, we describe their relation to Coxeter polytopes. The _Coxeter
diagram_ $\Sigma(P)$ of $P$ is a weighted graph with vertices corresponding to
facets $\\{f_{i}\\}_{i\in I}$, where two vertices corresponding to $f_{i}$ and
$f_{j}$ are connected by

  * •

an ordinary edge of weight $\cos(\frac{\pi}{k})$ if $f_{i}$ and $f_{j}$ meet
at angle $\frac{\pi}{k}$;

  * •

a dashed edge of weight $\cosh(\rho)$ if $f_{i}$ and $f_{j}$ diverge and are
at distance $\rho$ apart;

  * •

a bold edge of weight $1$ if $f_{i}$ and $f_{j}$ are parallel.

The _Gram matrix_ $M(P)$ of a Coxeter polytope $P$ is the Gram matrix of the
corresponding Coxeter diagram, i.e., $M(\Sigma(P))$. Note that $M(P)$
coincides with the Gram matrix of unit normal vectors to the facets of $P$.

Recall that the _signature_ of a real symmetric matrix is a triple
$(n_{+},n_{-},n_{0})$ of non-negative integers where $n_{+}$ is the number of
positive eigenvalues (the _positive inertia index_), $n_{-}$ is the number of
negative eigenvalues (the _negative inertia index_), and $n_{0}$ is the
multiplicity of $0$ as an eigenvalue, i.e., the nullity. Thus, a real
symmetric matrix is _positive definite_ if it has signature $(n,0,0)$. A
matrix is called _indecomposable_ if it cannot be transformed into a block-
diagonal matrix via simultaneous permutations of columns and rows.

An abstract Coxeter diagram $\Sigma$ is called

  * •

_elliptic_ if $M(\Sigma)$ is positive definite (the connected elliptic
diagrams are listed in Figure 2);

  * •

_parabolic_ if any indecomposable component of $M(\Sigma)$ is singular, and
every proper subdiagram of an indecomposable component is elliptic;

  * •

_Lannér_ if $\Sigma$ is connected and is neither elliptic nor parabolic, and
any proper subdiagram of $\Sigma$ is elliptic (all Lannér diagrams are listed
in Figure 3);

  * •

_hyperbolic_ if $M(\Sigma)$ has negative inertia index $n_{-}=1$;

  * •

_superhyperbolic_ if $M(\Sigma)$ has negative inertia index $n_{-}>1$;

  * •

_admissible_ , following the terminology from [28], if $M(\Sigma)$ is not
superhyperbolic and contains no parabolic subdiagrams.

Figure 2. The list of all connected elliptic diagrams along with their type,
where the subscript corresponds to the number of vertices. Figure 3. The list
of all Lannér diagrams, classified by Lannér in [22, Table 3].

######  Remark 4.3.

In [29], Vinberg shows that if $\Sigma=\Sigma(P)$ is the Coxeter diagram of a
compact hyperbolic $d$-polytope $P$, then $\Sigma$ is an admissible connected
hyperbolic diagram with positive inertia index $d$. Note that this implies
that $\Sigma$ does not contain any parabolic subdiagrams nor bold edges.

We can see from the above remark that $M(\Sigma)$ has rank $d+1$, so any
principal minor of order $d+2$ or larger is singular. Though we often omit the
weights of dashed edges when drawing Coxeter diagrams, these can be determined
from the remaining weights by applying this rank condition. The face structure
of $\Sigma(P)$ can be determined by the following proposition.

######  Proposition 4.4 ([29, Proposition 3.1]).

Let $P=\bigcap_{i\in I}f^{-}_{i}\subset\mathbb{H}^{d}$ be an acute-angled
polytope and $M$ its Gram matrix. Let $J\subset I$ be a subset such that the
induced submatrix $M_{J}$ is positive semidefinite. Then
$P^{J}=P\cap\left(\bigcap_{i\in J}f_{i}\right)$ is a face of $P$ if and only
if $M_{J}$ is positive definite. If so, $P^{J}$ is a face of codimension
$|J|$.

Thus, the faces of codimension $m$ in $P$ correspond to the elliptic
subdiagrams of order $m$ in $\Sigma(P)$, and hence the missing faces of $P$
correspond to the Lannér subdiagrams of $\Sigma(P)$. In particular, a compact
hyperbolic polytope has no missing faces of order greater than $5$. Thus,
using the data of the positive semidefinite submatrices of a Gram matrix, one
can determine the combinatorial type of the corresponding polytope.

######  Remark 4.5.

Note that Proposition 4.4 implies that every compact hyperbolic Coxeter
$d$-polytope is simple, i.e., every $(d-j)$-face is contained in precisely $j$
facets.

###  4.1. Polytope faces and Coxeter subdiagrams

Given a face of a compact Coxeter polytope, we discuss restrictions on the
combinatorial structure of the face and, if the face is a Coxeter polytope,
the structure of its Coxeter diagram. Let $P$ be a compact Coxeter
$d$-polytope with Coxeter diagram $\Sigma=\Sigma(P)$, and $S_{0}$ an elliptic
subdiagram of order $m$. By Proposition 4.4, $S_{0}$ corresponds to a face of
codimension $m$ in $P$; denote this face by $P(S_{0})$. While $P(S_{0})$ is
necessarily an acute-angled polytope, it may not be Coxeter in certain cases.
Borcherds provided the following sufficient condition for $P(S_{0})$ to be
Coxeter.

######  Proposition 4.6 ([6, Example 5.6]).

Suppose $P$ is a Coxeter polytope with diagram $\Sigma$, and
$S_{0}\subset\Sigma$ is an elliptic subdiagram containing no connected
component of type $A_{n}$ or $D_{5}$. Then $P(S_{0})$ is a Coxeter polytope.

We now examine what can be said about the Coxeter diagram of $P(S_{0})$ for
$S_{0}$ satisfying the conditions of Proposition 4.6. Supposing that
$P(S_{0})$ is indeed Coxeter, denote its Coxeter diagram by $\Sigma_{S_{0}}$.
The following terminology was introduced in [12, Section 1], which contains
further details of the computation of the dihedral angles in $P(S_{0})$. A
vertex of $\Sigma$ _attaches_ to $S_{0}$ if it is joined with any vertex of
$S_{0}$ by an edge of any type, in which case $w$ is called a _neighbour_ of
$S_{0}$. Then $w$ is called a _good neighbour_ if $\langle S_{0},w\rangle$ is
elliptic, and _bad_ otherwise. Note that in the latter case $\langle
S_{0},w\rangle$ must contain a Lannér diagram.

Let $\overline{S_{0}}$ denote the subdiagram of $\Sigma$ induced by the
vertices corresponding to facets of $P(S_{0})$, i.e., the good neighbours of
$S_{0}$ along with all vertices not attached to $S_{0}$. We now use the
following results of Felikson and Tumarkin, based on the analysis of Allcock
[4, Theorem 2.2], to describe the possible differences between the diagrams
$\Sigma_{S_{0}}$ and $\overline{S_{0}}$. A _simple edge_ refers to an ordinary
edge of weight $\cos\left(\frac{\pi}{3}\right)$, denoted by an ordinary
unlabelled edge of multiplicity $1$ in a Coxeter diagram.

######  Proposition 4.7 ([12, Corollary 1.1]).

Under the hypotheses of Proposition 4.6,

  1. (a)

If $S_{0}$ is of the type $H_{4}$, $F_{4}$, or $G_{2}^{(m)}$ for $m\geq 6$, or
any other diagram having no good neighbours, then
$\overline{S_{0}}=\Sigma_{S_{0}}$.

  2. (b)

If $S_{0}$ is of the type $H_{3}$, then $\overline{S_{0}}$ may be obtained by
replacing some dashed edges by ordinary edges.

  3. (c)

If $S_{0}$ is of the type $G_{2}^{(5)}$, then $\overline{S_{0}}$ may be
obtained from $\Sigma_{S_{0}}$ by replacing some edges labelled by $10$ by
simple edges, and some dashed edges by ordinary edges.

  4. (d)

If $S_{0}$ is of the type $B_{n}$ for $n\geq 3$, then $\overline{S_{0}}$ may
be obtained from $\Sigma_{S_{0}}$ by replacing some double edges by simple
edges, and some dashed edges by ordinary edges.

  5. (e)

If $S_{0}$ is of the type $B_{2}=G_{2}^{(4)}$, then $\overline{S_{0}}$ may be
obtained from $\Sigma_{S_{0}}$ by replacing some double edges by simple edges,
and some dashed edges by ordinary or empty edges.

We make frequent use of these restrictions in the second portion of this
paper, when we bound the dimension of polytopes with few facets.

###  4.2. Local determinants

A technical tool that can help to identify superhyperbolic subdiagrams is the
local determinant, for which the theory was developed in [30]. Let $\Sigma$ be
an abstract Coxeter diagram, and let $T$ be a subdiagram of $\Sigma$ such that
$\det(\Sigma\operatorname{\setminus}T)\neq 0$. The _local determinant_ of
$\Sigma$ on the subdiagram $T$ is given by

$\det(\Sigma,T)=\frac{\det(\Sigma)}{\det(\Sigma\operatorname{\setminus}T)}\,.$

When $\Sigma$ is composed of two subdiagrams joined in a simple way, we have
the following two results to simplify the calculation of the local
determinant.

######  Proposition 4.8 ([30, Proposition 12]).

If a Coxeter diagram $\Sigma$ consists of two subdiagrams $\Sigma_{1}$ and
$\Sigma_{2}$ having a unique vertex $v$ in common, and if no vertex of
$\Sigma_{1}$ attaches to $\Sigma_{2}$, then

$\det(\Sigma,v)=\det(\Sigma_{1},v)+\det(\Sigma_{2},v)-1\,.$

######  Proposition 4.9 ([30, Proposition 13]).

If a Coxeter diagram $\Sigma$ is spanned by two disjoint subdiagrams
$\Sigma_{1}$ and $\Sigma_{2}$ joined by a unique edge $v_{1}v_{2}$ of weight
$a$, then

$\det(\Sigma,\langle
v_{1},v_{2}\rangle)=\det(\Sigma_{1},v_{1})\det(\Sigma_{2},v_{2})-a^{2}\,.$

In particular, under the hypotheses of Proposition 4.9, if $M(\Sigma)$ is
singular and $\Sigma\operatorname{\setminus}\\{v_{1},v_{2}\\}$ is elliptic,
then we must have $\det(\Sigma_{1},v_{1})\det(\Sigma_{2},v_{2})=a^{2}$. We are
particularly interested in the case when $\Sigma_{1}$ and $\Sigma_{2}$ are
Lannér triangles. It is straightforward to check that the local determinant of
a Lannér triangle (which is necessarily negative) increases in magnitude as a
function of its edge weights (see, e.g., [30]).

##  5\. Properties of Compact Hyperbolic $d$-Polytopes with $d+4$ Facets

Before delving into the characterisation of compact hyperbolic Coxeter
$d$-polytopes with $d+4$ facets, we first mention several restrictions on
these polytopes that we refer to throughout the classification.

The first restrictions we consider are related to the number of pairs of
disjoint facets. The compact Coxeter polytopes with no pairs of disjoint
facets are precisely the $d$-simplices, which have $d+1$ facets and were
classified by Lannér [22], along with the seven Esselmann polytopes, which are
$4$-polytopes with $6$ facets that were constructed by Esselmann in [11]; this
list was shown to be complete by Felikson and Tumarkin [13, Theorem A]. In
2009, Felikson and Tumarkin studied compact Coxeter polytopes with precisely
one pair of disjoint facets.

######  Theorem 5.1 ([14, Main Theorem]).

A compact hyperbolic Coxeter $d$-polytope with exactly one pair of non-
intersecting facets has at most $d+3$ facets.

Thus, in the setting of $d$-polytopes with $d+4$ facets, we can restrict to
looking at polytopes with at least two pairs of disjoint facets. This can be
strengthened in dimension $4$. Felikson and Tumarkin [15] later studied $d$
polytopes with $n$ facets having at most $n-d-2$ pairs of disjoint facets.
They gave a finite algorithm for listing these polytopes, and carried out this
algorithm in dimension $4$. The algorithm produced no previously-unknown
polytopes of dimension $4$, yielding the following theorem.

######  Theorem 5.2 ([15, Theorem 7.1]).

Any compact hyperbolic Coxeter $4$-polytope with $n$ facets having at most
$n-6$ pairs of disjoint facets satisfies $n\leq 7$.

For the special case $d=4$ and $n=8$, we immediately obtain the following
result.

######  Corollary 5.3.

Any compact hyperbolic Coxeter $4$-polytope with $8$ facets must have at least
three pairs of disjoint facets.

As we shall see in Section 6, there do exist such polytopes with exactly three
pairs of disjoint facets, in particular, those of combinatorial type $G_{10}$,
$G_{12}$, or $G_{14}$ (see Appendix A for their description).

Following the methods of Tumarkin in [28], we can use a geometric property of
polytopes to reduce our search. If a polytope has a facet $f$ that meets
precisely $d$ other facets, we call this facet a _prism facet_ , and the
corresponding vertex of the Coxeter diagram a _prism vertex_. Note that this
condition is equivalent to $f$ being a simplex, since as a $(d-1)$-polytope it
would have precisely $d$ facets. We additionally refer to all edges incident
to a prism vertex as _prism edges_. At each prism facet $f$, we can truncate
the polytope by a hyperplane $H$ to obtain a new polytope $P^{\prime}$ with
the following properties:

  * •

The new facet $f^{\prime}=P\cap H$ of $P^{\prime}$ does not intersect $f$,

  * •

$f^{\prime}$ is either disjoint from or orthogonal to each facet of
$P^{\prime}$, and

  * •

$P^{\prime}$ is combinatorially equivalent to $P$.

In other words, we can obtain $P$ from a combinatorially equivalent polytope
$P^{\prime}$, where all facets of $P^{\prime}$ corresponding to prism facets
of $P$ meet any incident facets at dihedral angle $\frac{\pi}{2}$, by gluing
Coxeter prisms onto $P^{\prime}$. The hyperplane $H$ can be obtained by
choosing successive facets of $f$ and transforming the bounding hyperplane
along each chosen facet of $f$ so that they meet at dihedral angle
$\frac{\pi}{2}$. This process terminates with $P^{\prime}$ being
combinatorially equivalent to $P$ since each transformation preserves the face
structure, as $P$ is acute-angled. In our classification, we first assume
prism facets already satisfy the dihedral angle condition, and then later
obtain any combinatorially equivalent polytopes by gluing compact Coxeter
prisms. The compact Coxeter prisms have been classified by Kaplinskaja [20].

Lastly, we mention a rank condition on the Gram matrix. As discussed in
Section 4, the Gram matrix of a compact Coxeter $d$-polytope has rank $d+1$.
This is because the normal vectors to the facets are in $\mathbb{H}^{d}$,
which can be embedded in $\mathbb{R}^{d+1}$. Thus, we have the following
condition on the principal minors.

######  Proposition 5.4.

If $P$ is a compact Coxeter $d$-polytope with $d+4$ facets, then every
subdiagram $\Sigma_{0}$ obtained by deleting two vertices of $\Sigma(P)$ has a
singular Gram matrix.

###  5.1. The set of multi-multiple edges

In this section, we mention several restrictions on which ordinary edges can
be multi-multiple, i.e., have weight $\cos\left(\frac{\pi}{m}\right)$ for
$m\geq 6$, in a compact Coxeter diagram. Restricting this set is critical to
making the classification in Sections 6 and 7 computationally feasible.

######  Lemma 5.5.

Let $v_{1}v_{2}$ be an ordinary edge in an admissible abstract Coxeter diagram
$\Sigma$. Suppose that

  1. (a)

$\Sigma$ has a Lannér diagram of order greater than $3$ containing $v_{1}$ and
$v_{2}$; or

  2. (b)

$\Sigma$ has no Lannér diagram containing $v_{1}$ and $v_{2}$, $\Sigma$ has a
Lannér diagram $L$ of order greater than $2$ containing $v_{2}$, and there is
no dashed edge from $v_{1}$ to any vertex of $L$ .

Then $v_{1}v_{2}$ has low weight.

###### Proof.

The first part is an immediate consequence of the characterisation of Lannér
diagrams given in Figure 3.

For the second part, note that $v_{2}$ must be incident to an ordinary edge
induced by $L$; call this edge $v_{2}v_{3}$. If $v_{1}v_{2}$ was multi-
multiple, then $\\{v_{1},v_{2},v_{3}\\}$ would induce a parabolic diagram or
Lannér triangle, since $v_{1}v_{3}$ is not dashed by assumption. The former
cannot happen because $\Sigma$ is admissible, and the latter cannot happen
because we assume $\Sigma$ has no Lannér diagram containing both $v_{1}$ and
$v_{2}$. ∎

######  Lemma 5.6.

Suppose that $01234$ is a subdiagram of an admissible abstract Coxeter
diagram. If the induced set of missing faces is $\\{012,234\\}$, then $02$,
$12$, $23$, and $24$ are not multi-multiple.

###### Proof.

Note that one of $23$ or $24$ must be non-empty in order for $234$ to be a
missing face of size $3$; without loss of generality suppose $23$ is non-
empty. Thus, if either $02$ or $12$ is multi-multiple, it would induce a
missing face $023$ or $123$, which is forbidden. Hence neither $02$ nor $12$
is multi-multiple; the analogous result for $23$ and $24$ can be obtained by
swapping the roles of $0$,$1$ and $3$,$4$, respectively. ∎

######  Lemma 5.7 ([28, Lemma 4.14]).

There is no compact Coxeter $4$-polytope containing a subdiagram with induced
missing face list isomorphic to $\\{0123,014,235\\}$.

###  5.2. Admissible partial weightings

Now that we have discussed some general properties of compact Coxeter
$d$-polytopes with $d+4$ facets and the multi-multiple edges of their Coxeter
diagrams, we focus on the possible structure of the ordinary low-weight edges
of their Coxeter diagrams.

######  Definition 5.8.

We define the _partial low-weighting_ of an abstract Coxeter diagram $\Sigma$
as the image $\Sigma^{\leq 6}$ of a forgetful map $\varphi$ to a new weighted
diagram on the same vertex set where all edge weights of the form
$\cos\left(\frac{\pi}{m}\right)$ for $m\geq 6$ or $\cosh(\rho)$ for any
$\rho\in\mathbb{R}$ are forgotten, though the information of whether these
edges are ordinary or dashed remains. For a multi-multiple edge whose weight
was forgotten, we denote its new weight by $*$.

Figure 4 depicts the partial low-weighting of a Coxeter diagram, the same one
shown in Figure 1.

Figure 4. This illustrates the action of the forgetful map $\varphi$ on an
abstract Coxeter diagram $\Sigma$ (on the left) to obtain its partial low-
weighting (on the right). Note that the unique multi-multiple edge changes
weight from $10$ to $*$. Moreover, the dashed edges in the left diagram have
weight induced by Proposition 5.4, while the dashed edges on the right have no
associated weight.

Note that there are finitely many partial low-weightings of a given order, as
each edge is either simple, double, triple, dashed, or multi-multiple with
weight $*$. The forgetful map $\varphi$ also retains some information about
the elliptic subdiagrams: we say that a subdiagram $S_{0}$ of a partial low-
weighting is _elliptic_ if it remains elliptic when replacing the edges
weighted by $*$ with any weights of the form
$\cos\left(\frac{\pi}{m_{ij}}\right)$ for $m_{ij}\geq 6$. Since the only
connected elliptic diagrams containing a multi-multiple edge are $G_{2}^{(m)}$
for $m\geq 6$, this is well-defined. In particular, a subdiagram $S_{0}$ of a
partial low-weighting is elliptic if and only if any connected component of
$S_{0}$ is either comprised of ordinary edges and is elliptic, or consists of
a single edge with weight $*$. A _missing face_ of a partial low-weighting is
a set of vertices $S_{0}$ (corresponding to a set of facets) such that the
subdiagram $S_{0}$ is not elliptic, but every proper subdiagram is elliptic.

We say that a partial low-weighting $\Sigma_{1}^{\leq 6}$ _has the same
combinatorial type_ as (the partial low-weighting of) a Coxeter diagram
$\Sigma_{2}$ if their sets of missing faces are the same, up to a relabelling
of the vertices. We call a partial low-weighting on the vertex set
$\\{0,1,\dots,d+k-1\\}$ _admissible_ if it can be obtained as the image of an
admissible abstract Coxeter diagram. In particular, note that an _admissible_
partial low-weighting must not contain any parabolic subdiagrams.

Though we initially define partial low-weightings as the images of Coxeter
diagrams under the map $\varphi$, we can conversely consider constructing an
abstract Coxeter diagram from a low-partial weighting. We call an abstract
Coxeter diagram $\Sigma_{1}$ an _admissible extension_ of a partial low-
weighting $\Sigma_{2}^{\leq 6}$ if $\Sigma_{1}$ is admissible and
$\Sigma_{2}^{\leq 6}=\Sigma_{1}^{\leq 6}$.

A critical step in our method is ensuring that we have a system of polynomial
restrictions on our abstract Coxeter diagrams such that any valid partial low-
weighting is the image of at most one admissible Coxeter diagram satisfying
the polynomial restrictions. For each combinatorial type $G_{i}$, we denote
the corresponding system of equations by $V(G_{i})$. In our case, each system
is determined by checking that certain principal submatrices of $M(\Sigma)$ of
size at least $d+2$ are singular, which is satisfied for any abstract Coxeter
diagram corresponding to a $d$-polytope by Proposition 5.4. For an $n\times n$
matrix $M$ and $A,B\subseteq\\{0,1,\dots,n-1\\}$, let $M|_{A,B}$ denote the
submatrix formed by restricting to the rows with indices in $A$ and the
columns with indices in $B$ (where rows and columns are zero-indexed). Thus,
$M|_{A,B}$ is a $|A|\times|B|$ submatrix. The elements of $V(G_{i})$ are
represented by strings $i_{1}\cdots i_{m}$, representing the equation

$\det(M|_{S,S})=0\text{ where
}S=\\{0,\dots,n-1\\}\operatorname{\setminus}\\{i_{1},i_{2},\dots,i_{m}\\}\,.$

Note that while this conflicts with our notation for subdiagrams/facets, these
strings are only ever written as members of a set $V(G)$ for some
combinatorial type $G$, so the usage is made clear by the context.

######  Example 5.9.

Fix a $4\times 4$ matrix $M=(a_{i,j})_{0\leq i,j\leq 3}$. Then using our
notation, $V(G)=\\{01,13\\}$ is the system of equations

$\begin{cases}\det\left(\begin{bmatrix}a_{2,2}&a_{2,3}\\\
a_{3,2}&a_{3,3}\end{bmatrix}\right)=0\,,\\\ \\\
\det\left(\begin{bmatrix}a_{0,0}&a_{0,2}\\\
a_{2,0}&a_{2,2}\end{bmatrix}\right)=0\,,\end{cases}$

or, equivalently,

$\begin{cases}a_{2,2}a_{3,3}-a_{2,3}a_{3,2}=0\,,\\\
a_{0,0}a_{2,2}-a_{0,2}a_{2,0}=0\,.\end{cases}$

We now discuss our main method for classifying compact Coxeter polytopes,
which involves finding a sufficient set of restrictions such that all possible
compact Coxeter polytopes of a given combinatorial type can be listed by a
finite algorithm.

######  Proposition 5.10.

Fix a combinatorial type $G$, a set of restrictions $R(G)$ on the dihedral
angles, and a system of equations on the edge weights $V(G)$. Suppose $V(G)$
is chosen such that for every admissible partial low-weighting of type $G$
satisfying $R$, there are finitely many solutions to $V(G)$. Then all Coxeter
diagrams corresponding to compact polytopes of type $G$ satisfying $V(G)$ and
$R(G)$ can be listed by iterating over all admissible partial low-weightings
of $G$ and testing whether the solutions to $V(G)$ yield a diagram satisfying
the properties of Remark 4.3.

In our classification, we first fix a combinatorial type $G$ and then deduce
certain properties of the dihedral angles that must hold for any compact
Coxeter polytope of type $G$. This set of restrictions will comprise $R(G)$.
Moreover, our set of equations $V(G)$ will be a subset of the restrictions
guaranteeing that the Gram matrix of a $d$-polytope has rank $d+1$. Hence,
$V(G)$ is satisfied by the Gram matrix of any compact Coxeter $d$-polytope.
Imposing the conditions of $R(G)$ and $V(G)$ does not eliminate any compact
Coxeter polytopes of combinatorial type $G$. Thus, we obtain the following
corollary, stating that Proposition 5.10 yields all compact Coxeter polytopes
of type $G$ when $R(G)$ and $V(G)$ are as described.

######  Corollary 5.11.

Suppose $G$, $R(G)$, and $V(G)$ are chosen satisfying the hypotheses of
Proposition 5.10, and additionally that all equations in $V(G)$ come from
setting principal minors of size $\geq d+2$ to zero, and that the properties
in $R(G)$ are satisfied for all compact Coxeter polytopes of type $G$. Then
the process in Proposition 5.10 yields all Coxeter diagrams corresponding to
polytopes of type $G$.

###### Proof.

Since the rank of the Gram matrix $M(P)$ for any compact Coxeter $d$-polytope
$P$ is $d+1$, then every principal minor of size $d+2$ or larger must vanish.
Thus, all Coxeter diagrams corresponding to compact polytopes of type $G$
satisfy $V(G)$. By assumption, all compact Coxeter polytopes of type $G$ also
satisfy $R(G)$. The process in Proposition 5.10 yields the set of all compact
Coxeter polytopes of type $G$ satisfying $V(G)$ and $R(G)$, which is merely
equivalent to the set of all compact Coxeter polytopes of type $G$. ∎

We emphasize that in Proposition 5.10, our conditions on $V(G)$ require
iterating over all admissible partial low-weightings of type $G$ satisfying
$R(G)$. In practice, this involves checking that the system $V(G)$ has
finitely many solutions for each such low-weighting, which can be done with a
computer algebra system such as Mathematica. We often choose $V(G)$ to be
minimal or nearly minimal to satisfy the hypotheses of Proposition 5.10, since
in practice this speeds up the necessary computations. However, we are not
aware of a less computationally challenging method for checking the
sufficiency of the choice of $V(G)$, as the system may (and often does) have
infinitely many solutions when considering non-admissible partial low-
weightings, and enumerating the set of admissible partial low-weightings is
itself computationally complex.

###  5.3. Computational methods

In this section, we describe the computational steps used to classify compact
Coxeter polytopes of a given combinatorial type $G$, following the methods
described in Proposition 5.10 and Corollary 5.11. We begin with a set of $d+k$
vertices with unknown edge weights, and describe the process by which we
assign edge weights such that every admissible partial low-weighting of type
$G$ is constructed.

#### 1\. Select a set of multi-multiple edges

For each combinatorial type, we begin by restricting the set of edges which
can be multi-multiple according to the results in Subsection 5.1. In practice,
each combinatorial type considered in this paper can be limited to having at
most five multi-multiple edges. We then iterate over all subsets of the
possibly multi-multiple edges, fix such a subset $H$, and examine the Coxeter
diagrams for which the set of multi-multiple edges is precisely $H$.

#### 2\. Assign edge weights within Lannér diagrams of size $4$ and $5$

Recall that there are finitely many Lannér diagrams of size $4$ or $5$ (see
Figure 3). In particular, if we restrict an admissible partial low-weighting
to only those vertices contained in Lannér diagrams of size $4$ or $5$, the
resulting subdiagram is obtained by gluing Lannér diagrams from this list. We
can thus iterate over the possible Lannér diagrams and over permutations of
the vertices within each Lannér diagram to assign weightings to edges within
such a subdiagram.

#### 3\. Assign weights to the remaining ordinary low-weight edges

There are finitely many assignments of the remaining low-weight edges, i.e.,
those not contained within a Lannér diagram of size $4$ or $5$, since each
must have weight $0$, $1$, $2$, or $3$. We iterate over all these
possibilities, with the additional restriction that all subdiagrams not
containing a missing face must be elliptic (see Figure 2). We furthermore
require that subdiagram induced by any two Lannér diagrams is connected, lest
this subdiagram be superhyperbolic.

#### 4\. Solve for the remaining edge weights

At this stage, the only edge weights which have not been assigned are the
weights of the multi-multiple edges and dashed edges. The weights of the
multi-multiple edges must be real numbers in the range
$\left[\cos\left(\frac{\pi}{6}\right),1\right)=\left[\frac{\sqrt{3}}{2},1\right)$,
and the weights of the dashed edges must be real numbers in the range
$(1,\infty)$. Restricting to these ranges, we find all solutions to the system
of equations $V(G)$, where the unknowns are the weights of the multi-multiple
and dashed edges. We do so using the computer algebra system Mathematica. By
assumption, $V(G)$ is chosen so that this solution set is finite.

#### 5\. Check the signatures of the resulting matrices.

For each of the solutions to the system of equations, we obtain a potential
Gram matrix of a compact Coxeter polytope. By Remark 4.3, it is sufficient to
check whether the signature of the resulting matrix is $(d,1,k-1)$. If so,
this is the Gram matrix of a compact Coxeter $d$-polytope with $d+k$ facets of
combinatorial type $G$.

#### 6\. Glue prisms onto any prism facets

Recall that we initially assumed the weights of all prism edges were $0$. If
any polytopes are constructed by the process above, we can obtain the complete
list of compact Coxeter polytopes of combinatorial type $G$ by gluing compact
Coxeter prisms to this prism facet. There are finitely many compact Coxeter
prisms of dimension $4$ or $5$, see [20] for a complete list.

The code implementing the process described above is publicly available at
https://github.com/agburcroff/Cox_d-Polytopes_with_dplus4_Facets.git.

##  6\. Classification of Compact Coxeter $4$-Polytopes with $8$ Facets

In this section, we explain how to determine all compact Coxeter $4$-polytopes
with $8$ facets. We handle each of the $34$ possible combinatorial types
determined in Section 3 and listed in Appendix A individually.

For each such combinatorial type, we prove a sufficient number of restrictions
on the Coxeter diagram of any polytope realising this type so that the
possible Coxeter diagrams can be listed by a finite algorithm. To be more
specific, we claim that given any possible assignment of weights on the
ordinary edges of $\Sigma$, there are a finite number of solutions for the
weights of the remaining edges given the system of equations for
$\operatorname{rank}(M(\Sigma))$ i.e, that every principal minor of order $6$
in $M(\Sigma)$ has determinant $0$.

Given a combinatorial type, the task of enumerating all compact polytopes with
this type is a rather involved task. Due to the presence of multi-multiple
edges, initially there is an infinite number of potential Gram matrices that
must be searched. In this section, we detail how to use information about the
elliptic and Lannér subdiagrams to reduce the number of multi-multiple edges,
then, once we have reduced to a system of algebraic equations with finitely
many solutions, computationally determine those solutions which yield compact
polytopes.

We first eliminate several combinatorial types by applying the previous work
of Felikson and Tumarkin [13, 14, 15, 28].

######  Corollary 6.1.

There are no compact hyperbolic Coxeter $4$-polytopes of combinatorial types
$G_{31}$, $G_{32}$, $G_{33}$, or $G_{34}$.

###### Proof.

These combinatorial types all have exactly one pair of disjoint facets, and
thus cannot be realised by Theorem 5.1. ∎

Using Corollary 5.3 (following from results in [15]) we can immediately
exclude six more combinatorial types, each having precisely two pairs of
disjoint facets.

######  Corollary 6.2.

There are no compact hyperbolic Coxeter $4$-polytopes of combinatorial types
$G_{25}$, $G_{26}$, $G_{27}$, $G_{28}$, $G_{29}$, or $G_{30}$.

We can furthermore eliminate three combinatorial types by considering a
certain forbidden subdiagram.

######  Corollary 6.3.

There are no compact hyperbolic Coxeter $4$-polytopes of combinatorial types
$G_{22}$, $G_{23}$, or $G_{24}.$

###### Proof.

Each of these has a subdiagram with induced missing face list isomorphic to
$\\{0123,014,235\\}$, hence cannot be realised as a polytope by Lemma 5.7. ∎

It remains to determine whether there are polytopes of combinatorial type
$G_{i}$ for $i=1,2,\dots,21$, and if so, to classify these polytopes. We
handle each of these combinatorial types individually in the following
subsections. In each subsection, we prove a sufficient set of restrictions to
ensure that any polytopes realising a given combinatorial type can be listed
by a finite algorithm. One such initial restriction is that we can set the
weight of all prism edges to $0$ (see Section 5 for further explanation), and
then glue prisms to any polytopes obtained. In each case, the algorithm has
been successfully implemented via the program described in Subsection 5.3,
though this program can in some cases take several days to check one
combinatorial type on a standard machine.

### Combinatorial type $G_{1}$

Observe that this is the combinatorial type of a simplex which has been
truncated at three distinct vertices. In particular, vertices $v_{0}$,
$v_{1}$, and $v_{2}$, which correspond to the facets obtained by the
truncation, are all prism vertices. There are no possible multi-multiple edges
by Lemma 5.5, since all non-prism edges are contained in a Lannér diagram of
order $4$. This combinatorial type satisfies the conditions of Corollary 5.11
and can be handled via computer computation without further restriction using
the following system of equations:

$V(G_{1})=\\{01,02,04,12,13,23\\}\,.$

This yields $130$ compact Coxeter polytopes of type $G_{1}$.

### Combinatorial type $G_{2}$

This is the combinatorial type of a simplex that has been truncated at two
vertices and one edge, disjoint from the two vertices. The vertices $v_{0}$
and $v_{1}$ are both prism vertices. There are no possible multi-multiple
edges, as all non-prism edges satisfy the hypotheses of Lemma 5.5. Similarly
to the previous case, this combinatorial type satisfies the conditions of
Corollary 5.11 with:

$V(G_{2})=\\{01,03,04,12,13,23\\}\,.$

This yields $115$ compact Coxeter polytopes of type $G_{2}$.

### Combinatorial type $G_{3}$

This combinatorial type, like the previous two, is also relatively
straightforward to classify. Note that the prism vertices are $v_{0}$ and
$v_{3}$. There are no possible multi-multiple edges by Lemma 5.5. The system
of equations

$V(G_{3})=\\{02,03,07,12,13,23\\}$

provides sufficient restrictions to satisfy Corollary 5.11. This yields $49$
compact Coxeter polytopes of type $G_{3}$.

### Combinatorial type $G_{4}$

Since this combinatorial type consists of four pairs of disjoint facets, it is
the combinatorial type of a $4$-cube. The compact $4$-cubes were recently
classified by Jacquemet and Tschantz [19], which, similarly to the present
methods, utilised significant computer searches. There are $12$ such
polytopes.

### Combinatorial type $G_{5}$

This is the first example of a combinatorial type for which we need to apply
extra restrictions before passing to computer computation.

Note that there is at most one multi-multiple edge within the subdiagram
$567$, and one multi-multiple edge within the subdiagram $01234$. There are no
further possible multi-multiple edges. By symmetry, we can assume the multi-
multiple edges are limited to $57$ and $23$.

Suppose that both arise, say $57$ and $23$ are multi-multiple. Note then that
$46$, $06$, and $16$ are non-empty to ensure no two Lannér subdiagrams are
disjoint, and in fact they must be single edges. Moreover, $56$ and $57$ are
either both have multiplicity $1$, or $0$ and $1$, respectively. The only
other additional non-empty edges are $12$ and $03$, each with multiplicity at
most $3$. Up to symmetry, this leaves $2(1+3+9)=24$ diagrams to check. We can
do so using Corollary 5.11 and the system of equations

$V(G_{5})=\\{03,04,12,13,14,24,34\\}\,.$

Now suppose just $57$ is multi-multiple. The previous check actually
encompasses the case where precisely three vertices of $01234$ are adjacent to
$v_{6}$. Suppose there are four such vertices, say $0123$ are. Then all edges
between the subdiagram $0123$ and $v_{6}$ are simple edges, and $34$ and $45$
either both have multiplicity $1$, or $0$ and $1$, respectively. Then the only
additional edges are $02$ and $27$, each with multiplicity at most $3$. As in
the previous case, there are $24$ partial low-weightings to test, we can
complete this using Corollary 5.11 and the system of equations $V(G_{5})$
defined above. Lastly, suppose there are five such vertices, then we can
proceed similarly with just $2$ diagrams to check.

The remaining cases, namely when only $23$ is multi-multiple or when there are
no multi-multiple edges, can be handled with the same choice of $V(G_{5})$ as
above. This yields $3$ compact Coxeter polytopes of type $G_{5}$.

### Combinatorial type $G_{6}$

Note that $15$ and $35$ are not multi-multiple by applying Lemma 5.6 to the
subdiagram $13457$.

If $36$ is multi-multiple, then so is $37$. Suppose it is, and $37$ is not.
Then $46$, $56$, and $35$ are empty to prevent inducing a forbidden missing
face of size $3$. Then $57$ has multiplicity at least $2$ \- looking at the
Lannér subdiagrams, there are only two possibilities for the rank-$4$ Lannér
diagram (where in fact $57$ has multiplicity $3$). It can be checked that the
left triangle $145$ cannot be admissibly completed in either case.

Suppose $36$ and $37$ are multi-multiple. Then $4567$ is a Lannér path. We can
then use local determinants on $134567$, along with Proposition 4.9, to show
that there are no possibilities.

Otherwise, just $37$, $12$, $24$, and $14$ can be multi-multiple. Under this
assumption, $G_{6}$ satisfies the hypotheses of Corollary 5.11 with

$V(G_{6})=\\{01,02,06,12,14,16,23,26,27\\}\,.$

This yields $2$ compact Coxeter polytopes of type $G_{6}$.

### Combinatorial type $G_{7}$

Note that vertices $v_{0}$ and $v_{1}$ are both prism vertices. The only
possible multi-multiple edges are $34$ and one of $25$ or $26$ by Lemma 5.5.
Under these restrictions, we can apply Corollary 5.11 with

$V(G_{7})=\\{02,03,07,12,13,23\\}\,.$

This yields $2$ compact Coxeter polytopes of type $G_{7}$.

### Combinatorial type $G_{8}$

This has prism vertices $v_{0}$ and $v_{3}$, with the only possible multi-
multiple edges being $16$ and $27$ by Lemma 5.5. The system of equations

$V(G_{8})=\\{03,05,06,07,13,23,67\\}$

provides sufficient restrictions to satisfy Corollary 5.11. This yields $1$
compact Coxeter polytope of type $G_{8}$.

### Combinatorial type $G_{9}$

Note that the vertex $v_{0}$ is the only prism vertex. By Lemma 5.5, the only
possible multi-multiple edges are $36$, $37$, and $15$. Note that $36$ and
$37$ cannot simultaneously be multi-multiple, otherwise any assignment of
weights to the rank-$4$ Lannér diagram $4567$ induces a forbidden Lannér
triangle with $3$. By symmetry, we can assume $37$ is not multi-multiple.

Suppose $36$ and $15$ are multi-multiple. Note that $46$, $56$, and $57$ must
then be empty, hence the rank-$4$ Lannér diagram $4567$ is a path. For any
assignment of weights within this Lannér diagram, there is no additional non-
empty edge connecting Lannér diagram $25$ to the Lannér diagram $367$. Thus,
this cannot occur, as we cannot have two disjoint Lannér diagrams.

Suppose only edge $15$ is multi-multiple. The hypotheses of Corollary 5.11 are
satisfied with

$V(G_{9})=\\{01,04,05,12,13,15,45\\}\,.$

Suppose only edge $36$ is multi-multiple. We can again look at the structure
of the rank-$4$ Lannér diagram (noting $46$ and $56$ are empty) - if it is a
path, then we can repeat the same analysis as two paragraphs higher with
connecting $25$ to $367$. The only remaining option is that the rank-$4$
diagram consists of the star with one edge of multiplicity $3$ and two edges
of multiplicity $1$, with two possible positions (up to symmetry). Note that
the only additional edges are $15$ and $24$. We complete the analysis of this
case with $V(G_{9})$ as above.

Similarly, if all ordinary edges have low weight, then the same system of
equations $V(G_{9})$ suffices. This yields $15$ compact Coxeter polytopes of
type $G_{9}$.

### Combinatorial type $G_{10}$

By Lemma 5.6, the only possible multi-multiple edges are in the subdiagrams
$256$ and $034$. Note that not both $25$ and $26$ can be multi-multiple lest
$156$ cannot be admissibly completed, so assume $26$ has low weight. A similar
analysis can be used to show that at least one of $03$ or $04$ has low weight,
hence we can assume $04$ has low weight.

Suppose both $34$ and $56$ are multi-multiple. Note that we cannot have all
edges $13$, $14$, $15$, $16$ non-empty, less we form a $4$-cycle in the
subdiagram $134567$. This subdiagram cannot contain a $4$-cycle since it
contains no Lannér diagrams of size at least $4$, and all elliptic diagrams
are acyclic. Thus, we can assume by symmetry that the edge $13$ is empty.
Applying Proposition 4.9 to the subdiagram $013456$, we obtain that at least
one of $56$ or $34$ has multiplicity at most $13$. Under this assumption, we
can complete the analysis via Corollary 5.11 with

$V(G_{10})=\\{02,03,05,06,12,13,23\\}\,.$

Otherwise, assuming the multi-multiple edges are limited to $03$, $25$, and at
most one of $34$ or $56$, Corollary 5.11 similarly applies with $V(G_{10})$
defined as above. This yields $1$ compact Coxeter polytope of type $G_{10}$.

### Combinatorial type $G_{11}$

Observe that the only possible multi-multiple edges are those within the
subdiagrams $0167$ or $2345$.

Suppose that $45$ is multi-multiple. Observe that no other edges of $2345$ are
multi-multiple, lest $67$ cannot connect to one of the rank-$3$ Lannér
diagrams. Moreover, by symmetry we can assume the multi-multiple edges among
$0167$ are limited to $06$ and $17$ or $06$ and $16$. Suppose first that these
two edges are $06$ and $17$. We can apply Corollary 5.11 with

$V(G_{11})=\\{01,02,03,04,05,06,16,17,27,67\\}\,.$

The same analysis can be applied supposing that these two edges are $06$ and
$16$. Thus, we can now assume $45$ is not multi-multiple.

Suppose $06$ is multi-multiple. In order to ensure that $67$ is connected to
$245$ and $345$, we have three cases:

  1. (1)

Suppose $26$ and $37$ are non-empty. The only other possible multi-multiple
edge is $17$.

  2. (2)

Suppose $27$ and $37$ are non-empty. The only other possible multi-multiple
edge is $27$.

  3. (3)

Suppose $57$ is non-empty. The other possible multi-multiple edges are $24$,
$34$, and $16$. We cannot have $24$ and $34$ simultaneously multi-multiple,
lest we create a forbidden Lannér triangle, so assume by symmetry that $24$ is
not multi-multiple.

In all three cases, we can apply Corollary 5.11 with $V(G_{11})$ as above. We
can now assume the multi-multiple edges are among $1345$ excluding $34$, i.e.
either $13$, $45$ or $35$, $45$ or just one of these, and again the same
analysis applies. This yields $8$ compact Coxeter polytopes of type $G_{11}$.

### Combinatorial type $G_{12}$

Observe that $57$, $37$, $47$, $67$, and $27$ are not multi-multiple by Lemma
5.6.

Suppose $03$ is multi-multiple, and $34$ is not. Then $47$ is empty, and $34$
and $37$ have multiplicities $2,3$ or $3,3$, respectively. Thus, $67$ has
multiplicity $1$. Considering local determinants on $013457$, the only two
possibilities are that $03$ has multiplicity $4$, $34$ has multiplicity $2$ or
$3$, $37$ has multiplicity $3$, and $56$ has multiplicity $5$. Note that now
$03$ and $13$ must be empty. If edge $27$ is empty, we can consider local
determinants on $234567$ using Proposition 4.9 to see that we must have
multiplicity $2$ on $34$, and repeating local determinant analysis on $123467$
we find that the multiplicity of $26$ is $5$, $6$, or $7$. Then considering
local determinants on the subdiagram $234567$ yieldsthat $17$ is empty, and no
other can arise. This leaves us with three diagrams to check - none of these
yield polytopes. Now we can assume that $27$ is non-empty - in fact, it must
have multiplicity $1$. Considering local determinants on the subdiagram
$123467$ using Proposition 4.9 yields that the respective weights of $45$,
$34$, $67$ are $2,3,2$; $2,3,3$; $3,3,2$; or $3,2,2$. Moreover, the only other
possible non-empty edges are $25$ or $12$, each with multiplicity $1$. This
leaves us with $16$ diagrams to check - none of these yield polytopes.

Suppose $03$ and $34$ are multi-multiple. Then $07$ is empty. So we can
consider local determinants on $034567$, noting that if $37$ has weight larger
than $1$ then $67$ has multiplicity $1$. From this and the previous paragraph,
we can determine that there are no admissible diagrams with $03$ multi-
multiple.

We are now left with considering multi-multiple edges $34$, $56$, $26$, $25$.
We can then apply Corollary 5.11 with

$V(G_{12})=\\{05,12,15,16,25,26,35,56\\}\,.$

This yields $4$ compact Coxeter polytopes of type $G_{12}$.

### Combinatorial type $G_{13}$

Observe by Lemmas 5.5 and 5.6 that the only possible multi-multiple edges are
$12$, $56$, $16$, $25$, $34$, $03$, and $04$. We cannot simultaneously have
$03$ and $04$ multi-multiple lest $37$ and $47$ are both empty, so assume $04$
is not. We cannot simultaneously have $16$ and $25$ multi-multiple lest $57$
and $47$ are both empty, so assume $25$ is not. Thus, we are limited to $12$,
$56$, $16$, $34$, and $03$ being multi-multiple.

Suppose $56$ and $03$ are multi-multiple. Then $37$ and $03$ are empty, so we
can consider local determinants on $034567$ using Proposition 4.9. Assume $57$
is non-empty. Note that either $67$ is non-empty, or $2$ is attached to $347$
to ensure $26$ is attached to $347$. If $67$ is non-empty, the local
determinant on $034567$ with respect to $07$ cannot be zero, which is a
contradiction. If $2$ is attached to $7$ or $4$, then $47$ cannot have
multiplicity $3$ so is $2$. Considering the same local determinant shows that
this is not possible. So $23$ is non-empty, and by local determinants we must
have that $03$ has multiplicity $4$ or $5$, $34$ has multiplicity $2$ or $3$,
$47$ has multiplicity $3$, and both $57$ and $23$ have multiplicity $1$, with
no other edges being nonempty. Now take the determinant of the subdiagram
$034567$; this is non-zero, a contradiction.

Suppose $34$ and $03$ are multi-multiple (we can assume $56$ is not from
previous paragraph). Then $37$ is empty, so $47$ is not. We can now use local
determinants on $034567$. From this, we determine that $567$ must have one
empty edge. Thus, one of $67$, $57$ has multiplicity at least $2$, so $47$ has
multiplicity $1$. Repeating a similar local determinant argument, we find that
there are no admissible diagrams of this type.

The remaining cases can be handled by Corollary 5.11 with

$V(G_{13})=\\{01,02,05,06,12,13,16,25,26\\}\,.$

This yields $4$ compact Coxeter polytopes of type $G_{13}$.

### Combinatorial type $G_{14}$

By similar considerations to type $G_{12}$, if any of $27$, $37$, $26$, $36$,
$02$, or $03$ is multi-multiple then so is $23$.

Suppose $27$ and $23$ are multi-multiple. Note that the other possible edges
(up to symmetry) can be limited to within $145$, $56$, $04$, $03$, and $36$.

  * •

Suppose $56$ and $04$ are non-empty. Then the only other possible multi-
multiple edge is $14$.

  * •

Suppose $56$ is non-empty, $04$ is empty. Then the only other possible multi-
multiple edges are $45$ and $14$.

  * •

Suppose $04$ is non-empty, $56$ is empty. Then the only other possible multi-
multiple edges are $15$ and $14$.

  * •

Suppose $56$ and $04$ are empty. Then the other possible multi-multiple edges
are $45$, $14$, and $15$. The only additional edges are $03$ and $36$, one of
which has multiplicity $1$ or $2$ and the other having multiplicity $1$.

In all these cases, we can apply Corollary 5.11 with

$V(G_{14})=\\{01,14,15,17,23,24,47,57\\}\,.$

We can now restrict the multi-multiple edges to $23$, $45$, $14$, and $15$,
and the same analysis applies along with the same choice of $V(G_{14})$. This
yields $2$ compact Coxeter polytopes of type $G_{14}$.

### Combinatorial type $G_{15}$

Look at the subdiagram generated by $234567$. It can quickly be checked that
the only possible multi-multiple edge is one of $26$ or $27$, suppose by
symmetry it is $26$. Then $36$ must be empty, so $37$ and $67$ must have
multiplicities $2$ and $3$ or both $3$. From this information, we can then
complete the remainder of the rank-$4$ Lannér diagram, keeping in mind that
$46$ must have multiplicity $0$ as well. After doing so, it can be easily
checked that the triangle $345$ cannot be completed to an admissible diagram.
Hence there are no compact Coxeter polytopes of this type.

### Combinatorial type $G_{16}$

Considering overlapping triangles, the only possible multi-multiple edges are
$15$, $12$, $03$, $04$, and $34$. We cannot have $03$ and $04$ simultaneously
multi-multiple, so assume $03$ has low weight. By similar considerations to
$G_{12}$, if $04$ is multi-multiple then so is $34$.

We can apply Corollary 5.11 with

$V(G_{16})=\\{01,02,07,12,17,24,25\\}$

to show that there are no compact Coxeter polytopes of this type.

### Combinatorial type $G_{17}$

By Lemma 5.6, the only possible multi-multiple edges are $36$, $37$, $14$, and
$15$. Note that $36$ and $37$ cannot be simultaneously multi-multiple; assume
by symmetry that $36$ has low weight. Moreover, $14$ and $15$ cannot
simultaneously be multi-multiple.

With these assumptions, we can apply Corollary 5.11 with

$V(G_{17})=\\{03,05,07,13,34,35,57\\}$

to show that there are no compact Coxeter polytopes of this type.

### Combinatorial type $G_{18}$

Considering overlapping triangles, the only possible multi-multiple edges are
$12$, $37$, $36$, $25$, and $14$.

Suppose all five of these edges are multi-multiple. Then the rank-$4$ Lannér
diagram must be a path, and there are no additional non-empty edges. We can
check that there are no solutions to the determinants of the subdiagrams
$123456$ and $012345$ being zero, assuming that the undetermined edges have
weight in the range $\left[-\cos\left(\frac{\pi}{6}\right),0\right)$. Thus, we
can assume at most four of these edges are multi-multiple. We can now complete
the analysis with Corollary 5.11, setting

$V(G_{18})=\\{01,02,03,04,05,12,23,13\\}\,,$

to show that there are no compact Coxeter polytopes of this type.

### Combinatorial type $G_{19}$

By Lemma 5.6, the only possible multi-multiple edges are $15$, $14$, $35$,
$37$, $24$, and $26$. Note $35$ and $15$ cannot simultaneously be multi-
multiple, and similarly for $24$ and $14$. We can now complete the analysis
with Corollary 5.11, setting

$V(G_{19})=\\{02,03,05,06,07,12,13,23\\}\,,$

to show that there are no compact Coxeter polytopes of this type.

### Combinatorial type $G_{20}$

By Lemma 5.6, the only possible multi-multiple edges are contained in the
subdiagram $167$. We can then apply Corollary 5.11 with

$V(G_{20})=\\{05,06,12,15,16,56\\}\,.$

This shows that there are no compact Coxeter polytopes of this type.

### Combinatorial type $G_{21}$

By Corollary 5.11, the only possible multi-multiple edges are $05$, $07$,
$15$, $13$, $34$, $46$, $26$, and $27$. Note that $05$ and $07$ cannot
simultaneously be multi-multiple, lest $57$ cannot connect to $346$. By
symmetry, assume $07$ is not multi-multiple. We can then apply Corollary 5.11
with

$V(G_{21})=\\{01,02,03,04,05,06,07,12,15,25\\}\,,$

to show that there are no compact Coxeter polytopes of this type.

##  7\. Classification of Compact Coxeter $5$-Polytopes with $9$ Facets

We now proceed to the classification of compact Coxeter $5$-polytopes with $9$
facets. Though the process detailed in Section 3 yields more possible
combinatorial types in dimension $5$ than in dimension $4$, only six of these
($H_{i}$ for $1\leq i\leq 6$) are realised by compact Coxeter polytopes.
Moreover, these combinatorial types in general have a more restrictive face
structure than in the lower-dimensional cases, thus in most cases find a set
of equations for which the hypotheses of Corollary 5.11 are satisfied without
first proving additional restrictions. The main challenges are limiting the
set of multi-multiple edges and determining the set of equations $V(H)$ for
each combinatorial type $H$. We examine this process in detail for a few
combinatorial types, but we often omit the argument for routine cases. In
these cases, Lemmas 5.5 and 5.6 are sufficient to limit the multi-multiple
edges, and of course the claimed properties of $V(H)$ are checked
computationally.

#### Combinatorial type $H_{1}$

Note that $v_{0}$ and $v_{1}$ are both prism vertices. By [10, Lemma 5.3], it
is easily checked that there are four possible subdiagrams with the proper
missing face structure on $2345678$. We can thus obtain all diagrams by taking
these four subdiagrams and gluing prisms onto facets $0$ and $1$. This process
yields $22$ compact polytopes of type $H_{1}$.

#### Combinatorial type $H_{2}$

Note that $v_{0}$ and $v_{3}$ are both prism vertices. It is straightforward
to check that there are only four possible subdiagrams for $1245678$ by gluing
together subdiagrams from [10, Lemma 5.3]. We thus obtain all $18$ compact
polytopes of type $H_{2}$ by gluing prisms as appropriate to these
subdiagrams.

#### Combinatorial type $H_{3}$

At most one of the edges $25$ or $26$ can be multi-multiple. By symmetry, we
can assume that only $25$ can be multi-multiple. We can then apply Corollary
5.11 with $V(H_{3})=\\{23,15,12,04,02,01,13,15\\}$. This yields $6$ compact
Coxeter polytopes of type $H_{3}$.

#### Combinatorial type $H_{4}$

Only possible multi-multiple edges are $37$ or $38$. These can not be multi-
multiple simultaneously. We can then take
$V(H_{4})=\\{02,03,07,08,23,27,38\\}$. This yields $3$ compact Coxeter
polytopes of type $H_{4}$.

#### Combinatorial type $H_{5}$

The possible multi-multiple edges are restricted to $25$, $04$, $03$, $14$,
and $13$, some of which cannot be multi-multiple simultaneously. By symmetry
between $3$ and $4$, we can restrict to $25$, $04$, and $14$ possibly being
multi-multiple. We then take $V(H_{5})=\\{01,03,05,12,14,45\\}$. This yields
$1$ compact Coxeter polytope of type $H_{5}$.

#### Combinatorial type $H_{6}$

There are no possible multi-multiple edges. We can take
$V(H_{6})=\\{01,02,04,12,13,23\\}$. This yields $1$ compact Coxeter polytope
of type $H_{6}$.

#### Combinatorial type $H_{7}$

At most one of the edges $27$ or $28$ can be multi-multiple. By symmetry, we
can assume that only $28$ can be multi-multiple. We can take
$V(H_{7})=\\{02,03,12,13,23\\}$.

#### Combinatorial type $H_{8}$

The only possible multi-multiple edge is $28$. We can take
$V(H_{8})=\\{02,03,08,13,23,27,28,38\\}$.

#### Combinatorial type $H_{9}$

The only possible multi-multiple edge is $34$. We can take
$V(H_{9})=\\{01,02,03,04,12,13,14,23\\}$.

#### Combinatorial type $H_{10}$

The only possible multi-multiple edge is $12$. We can take
$V(H_{10})=\\{01,02,03,12,18,28\\}$.

#### Combinatorial type $H_{11}$

The only possible multi-multiple edges are $36$ or $37$, but not both. As
these cannot be multi-multiple simultaneously, we can assume by symmetry that
$37$ has low weight. We can take $V(H_{11})=\\{02,03,07,08,12,23,28,38\\}$.

#### Combinatorial type $H_{12}$

The only possible multi-multiple edges either $04$ and $14$, or $03$ and $13$.
By symmetry, we can assume the possible multi-multiple edges are only $03$ and
$13$. We can take $V(H_{12})=\\{01,02,03,05,12,13,35\\}$.

#### Combinatorial type $H_{13}$

The only possible multi-multiple edges are one of $26$ or $27$, and one of
$14$ or $15$. By symmetry, we can assume the multi-multiple edges are limited
to $14$ and $26$. We can take $V(H_{13})=\\{01,02,04,06,12,13,23\\}$.

#### Combinatorial type $H_{14}$

The only possible multi-multiple edges are one of $14$ or $15$. By symmetry,
we can assume $15$ has low weight. We can take
$V(H_{14})=\\{02,05,12,13,23,24\\}$.

#### Combinatorial type $H_{15}$

The only possible multi-multiple edges are one of $14$ or $15$. We can take
$V(H_{15})=\\{01,03,04,05,12,13,15,34,35\\}$.

#### Combinatorial type $H_{16}$

The multi-multiple edges limited to one of $03$ or $04$, and one of $13$ or
$15$. We can take $V(H_{16})=\\{01,03,05,06,12,13,16,36\\}$.

#### Combinatorial type $H_{17}$

The multi-multiple edges are limited to one of $03$ or $04$, and one of $27$
or $28$. By symmetry can limit ourselves to considering only $03$ and $27$. We
can take $V(H_{17})=\\{02,06,07,12,26,67\\}$.

#### Combinatorial type $H_{18}$

The multi-multiple edges are limited to $12$ and $14$. We can take
$V(H_{18})=\\{02,04,12,13,23\\}$.

#### Combinatorial type $H_{19}$

The multi-multiple edges are limited to $27$, $28$, $37$, and $38$. We can
take $V(H_{19})=\\{02,03,07,08,12,13,23\\}$.

#### Combinatorial type $H_{20}$

The multi-multiple edges are limited to $14$, $17$, $34$, and $37$. We can
take $V(H_{20})=\\{01,03,04,07,12,13,23\\}$.

#### Combinatorial type $H_{21}$

The only possible multi-multiple edge are $37$ or $38$, and by symmetry we can
assume $38$ has low weight. We can then take $V(H_{21})=\\{06,07,13,23,67\\}$.

#### Combinatorial type $H_{22}$

The only possible multi-multiple edges are $37$ and $38$. We can take
$V(H_{22})=\\{07,08,12,13,23\\}$.

#### Combinatorial type $H_{23}$

The only possible multi-multiple edges are $16$ and one of $37$ or $38$. By
symmetry, we can assume $38$ has low weight. We can take
$V(H_{23})=\\{01,03,06,12,13,23\\}$.

#### Combinatorial type $H_{24}$

The only possible multi-multiple edges are $24$ and $27$. We can take
$V(H_{24})=\\{04,07,12,13,23\\}$.

#### Combinatorial type $H_{25}$

The only possible multi-multiple edges are $02$, $05$, $17$. We can take
$V(H_{25})=\\{01,02,12,14,17,23,47\\}$.

#### Combinatorial type $H_{26}$

The only possible multi-multiple edges are $12$, $14$, $15$, and $24$. We can
take $V(H_{26})=\\{01,02,04,12,14,16,23,25\\}$.

#### Combinatorial type $H_{27}$

The only possible multi-multiple edges are $12$, $24$, and $25$. We can take
$V(H_{27})=\\{04,05,12,14,23,45\\}$.

#### Combinatorial type $H_{28}$

The only possible multi-multiple edges are $14$, $15$, and $24$. We can take
$V(H_{28})=\\{01,02,04,12,13,23\\}$.

#### Combinatorial type $H_{29}$

The only possible multi-multiple edges are $14$, $24$, and $27$. We can take
$V(H_{29})=\\{01,02,04,12,13,23\\}$.

#### Combinatorial type $H_{30}$

The only possible multi-multiple edge is $14$. We can take
$V(H_{30})=\\{05,12,13,23\\}$.

#### Combinatorial type $H_{31}$

The only possible multi-multiple edge is $38$. We can take
$V(H_{31})=\\{07,12,13,23\\}$.

#### Combinatorial type $H_{32}$

There are no possible multi-multiple edges. We can take
$V(H_{32})=\\{12,13,23\\}$.

#### Combinatorial type $H_{33}$

The only possible multi-multiple edges are $15$ or $24$; by symmetry we can
assume $24$ has low weight. We can take $V(H_{33})=\\{01,02,05,12,13,23\\}$.

#### Combinatorial type $H_{34}$

The only possible multi-multiple edges are one of $25$ or $26$, and one of
$37$ or $38$. By symmetry, we can assume $26$ and $38$ have low weight. We can
take $V(H_{34})=\\{02,05,07,12,13,23\\}$.

#### Combinatorial type $H_{35}$

The only possible multi-multiple edges are $25$, $26$, and one of $03$ or
$04$. By symmetry, we can assume $04$ has low weight. We can take
$V(H_{35})=\\{02,05,06,12,25,56\\}$.

#### Combinatorial type $H_{36}$

The only possible multi-multiple edges are $34$ and one of $03$ or $04$. By
symmetry, we can assume $04$ has low weight. We can take
$V(H_{36})=\\{07,12,13,23\\}$.

#### Combinatorial type $H_{37}$

The only possible multi-multiple edges are $27$ and one of $03$ or $04$. By
symmetry, we can assume $04$ has low weight. We can take
$V(H_{37})=\\{08,12,13,23\\}$.

#### Combinatorial type $H_{38}$

The only possible multi-multiple edges are $07$, $08$, $17$, $18$. We can take
$V(H_{38})=\\{01,03,07,08,12,17,18,27\\}$.

#### Combinatorial type $H_{39}$

The only possible multi-multiple edges are $34$ and one of $03$ or $04$. By
symmetry, we can assume $04$ has low weight. We can take
$V(H_{39})=\\{08,12,13,23\\}$.

#### Combinatorial type $H_{40}$

The only possible multi-multiple edges are $15$ and one of $03$ or $04$. By
symmetry, we can assume $04$ has low weight. We can take
$V(H_{40})=\\{06,12,13,23\\}$.

#### Combinatorial type $H_{41}$

The only possible multi-multiple edges are $03$, $04$, and $15$. We can take
$V(H_{41})=\\{06,12,13,14,23\\}$.

#### Combinatorial type $H_{42}$

The only possible multi-multiple edges are $02$, $03$, and $23$. We can take
$V(H_{42})=\\{03,07,12,13,23\\}$.

#### Combinatorial type $H_{43}$

The only possible multi-multiple edges are $03$ and $34$. We can take
$V(H_{43})=\\{01,04,13,14,23\\}$.

#### Combinatorial type $H_{44}$

The only possible multi-multiple edges are $12$, $15$, $25$, and one of $03$
or $04$. By symmetry, we can assume $04$ has low weight. We can take
$V(H_{44})=\\{02,05,07,12,17,23,57\\}$.

#### Combinatorial type $H_{45}$

The only possible multi-multiple edges are $03$, $04$, $13$, $15$, and $34$.
We can take $V(H_{45})=\\{01,03,05,12,14,23,24\\}$.

#### Combinatorial type $H_{46}$

The only possible multi-multiple edges are $04$, $15$, and one of $03$ or
$13$. By symmetry, we can assume $13$ has low weight. We can take
$V(H_{46})=\\{05,06,12,13,14,24\\}$.

#### Combinatorial type $H_{47}$

The only possible multi-multiple edges are $03$, $04$, $13$, $15$, and $34$.
We can take $V(H_{47})=\\{01,03,04,12,14,23,24\\}$.

#### Combinatorial type $H_{48}$

The only possible multi-multiple edges are $03$, $04$, and $34$. We can take
$V(H_{48})=\\{05,12,13,14,23\\}$.

#### Combinatorial type $H_{49}$

The only possible multi-multiple edges are $27$, $28$, and $78$. We can take
$V(H_{49})=\\{07,08,12,27,28,78\\}$.

#### Combinatorial type $H_{50}$

The only possible multi-multiple edges are $34$ and one of $03$ or $04$. By
symmetry, we can assume $04$ has low weight. We can take
$V(H_{50})=\\{02,04,12,13,23\\}$.

#### Combinatorial type $H_{51}$

The only possible multi-multiple edges are $26$, $27$, and $67$. We can take
$V(H_{51})=\\{06,07,12,26,27,67\\}$.

#### Combinatorial type $H_{52}$

The only possible multi-multiple edges are $27$, $28$, $67$. We can take
$V(H_{52})=\\{02,07,08,27,78\\}$.

#### Combinatorial type $H_{53}$

The only possible multi-multiple edges are $17$, $18$, $28$, and $78$. We can
take $V(H_{53})=\\{07,08,12,17,18,78\\}$.

#### Combinatorial type $H_{54}$

The only possible multi-multiple edges are $16$, $27$, and $67$. We can take
$V(H_{54})=\\{02,06,07,17,26\\}$.

#### Combinatorial type $H_{55}$

The only possible multi-multiple edges are $34$ and one of $03$ or $04$. By
symmetry, we can assume $04$ has low weight. We can take
$V(H_{55})=\\{08,12,13,23\\}$.

#### Combinatorial type $H_{56}$

The only possible multi-multiple edges are $13$, $16$, and $67$. We can take
$V(H_{56})=\\{01,03,06,16,36\\}$.

#### Combinatorial type $H_{57}$

The only possible multi-multiple edges are $07$, $26$, $27$, and one of $03$
or $04$. By symmetry, we can assume $04$ has low weight. We can take
$V(H_{57})=\\{02,06,07,17,23,27,37\\}$.

#### Combinatorial type $H_{58}$

The only possible multi-multiple edges are $25$, $28$, and $68$. We can take
$V(H_{58})=\\{05,08,12,28,58\\}$.

#### Combinatorial type $H_{59}$

The only possible multi-multiple edges are $03$, $04$, $15$, $16$. We can take
$V(H_{59})=\\{01,05,06,13,14,24,34\\}$.

#### Combinatorial type $H_{60}$

The only possible multi-multiple edges are $02$, $03$, $13$, $15$. We can take
$V(H_{60})=\\{01,03,08,13,18,23\\}$.

#### Combinatorial type $H_{61}$

The only possible multi-multiple edges are $12$, $15$, $26$, and one of $03$
or $04$. By symmetry, we can assume $04$ has low weight. We can take
$V(H_{61})=\\{01,02,06,12,13,23\\}$.

#### Combinatorial type $H_{62}$

The only possible multi-multiple edges are $02$, $04$, and $24$. We can take
$V(H_{62})=\\{02,07,14,24,27\\}$.

#### Combinatorial type $H_{63}$

The only possible multi-multiple edges are $03$ and $04$. We can take
$V(H_{63})=\\{12,13,14,24\\}$.

#### Combinatorial type $H_{64}$

The only possible multi-multiple edges are $67$ and one of $17$ or $26$. By
symmetry, we can assume $26$ has low weight. We can take
$V(H_{64})=\\{06,07,12,16,17,67\\}$.

#### Combinatorial type $H_{65}$

The only possible multi-multiple edge is $67$. We can take
$V(H_{65})=\\{01,17,26\\}$.

#### Combinatorial type $H_{66}$

The only possible multi-multiple edges are $02$, $04$, $14$, $23$, and one of
$15$ or $16$. By symmetry, we can assume $16$ has low weight. We can take
$V(H_{66})=\\{01,04,08,12,14,15,24\\}$.

#### Combinatorial type $H_{67}$

The only possible multi-multiple edges are $13$ and $35$. We can take
$V(H_{67})=\\{02,12,13,23\\}$.

#### Combinatorial type $H_{68}$

The only possible multi-multiple edges are $34$, $03$, $13$, $04$, and $14$.
Note that if $03$ is multi-multiple, then $04$ and $14$ have low weight, lest
a forbidden Lannér diagram be induced among $01348$. Similar arguments hold
when swapping the roles of vertices $3,4$ or $0,1$, so by symmetry we can
assume $04$ and $14$ have low weight. Then we can take
$V(H_{68})=\\{01,03,05,12,13,23\\}$.

#### Combinatorial type $H_{69}$

The only possible multi-multiple edges are $03$, $04$, $13$, and $15$. We can
take $V(H_{69})=\\{03,05,12,13,14,23\\}$.

#### Combinatorial type $H_{70}$

The only possible multi-multiple edges are $03$, $04$, $15$, and $28$. We can
take $V(H_{70})=\\{05,08,12,13,23,24\\}$.

#### Combinatorial type $H_{71}$

The only possible multi-multiple edges are $03$, $04$, and one of $12$ or
$25$. By symmetry, we can assume $25$ has low weight. We can take
$V(H_{71})=\\{01,08,12,18,23,58\\}$.

#### Combinatorial type $H_{72}$

The only possible multi-multiple edges are $03$, $13$, and $15$. We can take
$V(H_{72})=\\{03,05,12,13,23\\}$.

#### Combinatorial type $H_{73}$

The possible multi-multiple edges are $03$, $04$, $13$, $15$, $67$. We can
take $V(H_{73})=\\{01,03,06,12,13,14,35,36\\}$.

#### Combinatorial type $H_{74}$

The only possible multi-multiple edges are $16$, $27$, and one of $03$ or
$04$. By symmetry, we can assume $04$ has low weight. We can take
$V(H_{74})=\\{06,07,12,13,23\\}$.

#### Combinatorial type $H_{75}$

The only possible multi-multiple edges are $03$, $04$, $16$, and $27$. We can
take $V(H_{75})=\\{06,07,12,13,14,23\\}$.

#### Combinatorial type $H_{76}$

The only possible multi-multiple edges are $03$, $04$, $27$, and $28$. We can
take $V(H_{76})=\\{07,08,12,13,14,23\\}$.

#### Combinatorial type $H_{77}$

The only possible multi-multiple edges are one of $03$ or $04$, and one of
$26$ or $27$. By symmetry, we can assume $04$ and $26$ have low weight. We can
take $V(H_{77})=\\{06,08,12,13,23\\}$.

#### Combinatorial type $H_{78}$

The only possible multi-multiple edges are $26$, $27$, and one of $03$ or
$04$. By symmetry, we can assume $04$ has low weight. We can take
$V(H_{78})=\\{07,08,12,13,23\\}$.

#### Combinatorial type $H_{79}$

The only possible multi-multiple edges are $16$, one of $03$ or $04$, and one
of $27$ or $28$. By symmetry, we can assume $04$ and $28$ have low weight. We
can take $V(H_{79})=\\{02,06,07,12,13,23\\}$.

#### Combinatorial type $H_{80}$

The only possible multi-multiple edges are $02$, $03$, and $24$. We can take
$V(H_{80})=\\{04,08,12,13,23\\}$.

#### Combinatorial type $H_{81}$

The only possible multi-multiple edges are $16$, $26$, and $28$. We can take
$V(H_{81})=\\{02,06,08,26,68\\}$.

#### Combinatorial type $H_{82}$

The only possible multi-multiple edges are $02$, $04$, $15$, and $17$. We can
take $V(H_{82})=\\{01,05,07,12,14,24\\}$.

#### Combinatorial type $H_{83}$

The only possible multi-multiple edges are $03$, $04$, $15$, and $16$. We can
take $V(H_{83})=\\{05,06,12,12,13,24\\}$.

#### Combinatorial type $H_{84}$

The only possible multi-multiple edges are $02$, $04$, $15$, and $16$. We can
take $V(H_{84})=\\{01,04,06,12,14,24\\}$.

#### Combinatorial type $H_{85}$

The only possible multi-multiple edges are $02$, $03$, and $13$. We can take
$V(H_{85})=\\{01,08,12,13,23\\}$.

#### Combinatorial type $H_{86}$

The only possible multi-multiple edges are $03$, $04$, and $13$. We can take
$V(H_{86})=\\{08,12,13,14,23\\}$.

#### Combinatorial type $H_{87}$

The only possible multi-multiple edges are $04$, $17$, and one of $02$ or
$24$. By symmetry, we can assume $24$ has low weight. We can take
$V(H_{87})=\\{01,04,12,14,24\\}$.

#### Combinatorial type $H_{88}$

The only possible multi-multiple edges are $03$, $04$, $28$, and $34$. We can
take $V(H_{88})=\\{04,08,12,14,23,24\\}$.

#### Combinatorial type $H_{89}$

The only possible multi-multiple edges are $03$, $04$, $27$, and $28$. We can
take $V(H_{89})=\\{07,08,12,13,23,24\\}$.

#### Combinatorial type $H_{90}$

The only possible multi-multiple edges are $25$, $27$, and one of $03$ or
$04$. By symmetry, we can assume $04$ has low weight. We can take
$V(H_{90})=\\{05,07,12,13,23\\}$.

#### Combinatorial type $H_{91}$

The only possible multi-multiple edges are $02$, $04$, $16$, and $24$. We can
take $V(H_{91})=\\{02,04,06,14,24,26\\}$.

#### Combinatorial type $H_{92}$

The only possible multi-multiple edges are $03$, $04$, $28$, and $34$. We can
take $V(H_{92})=\\{03,08,12,14,23,24\\}$.

#### Combinatorial type $H_{93}$

The only possible multi-multiple edges are $03$, $04$, $15$, $16$, and $25$.
We can take $V(H_{93})=\\{01,05,06,12,13,14,23\\}$.

#### Combinatorial type $H_{94}$

The only possible multi-multiple edges are $05$, $13$, and $15$. We can take
$V(H_{94})=\\{01,05,08,15,18,25\\}$.

#### Combinatorial type $H_{95}$

The only possible multi-multiple edges are $04$, $08$, $14$, and $28$. We can
take $V(H_{95})=\\{01,04,08,14,24,28,48\\}$.

#### Combinatorial type $H_{96}$

The only possible multi-multiple edges are $02$, $04$, $23$, and $24$. We can
take $V(H_{96})=\\{02,04,08,12,14,24\\}$.

#### Combinatorial type $H_{97}$

The only possible multi-multiple edges are $03$, $04$, $13$, $24$, and one of
$16$, $17$, $26$, or $27$. By symmetry between vertices $6$ and $7$, we can
assume $17$, $26$, and $27$ have low weight. We can take
$V(H_{97})=\\{01,02,06,12,14,23,24\\}$.

#### Combinatorial type $H_{98}$

The only possible multi-multiple edge is one of $03$ or $04$; by symmetry, we
can assume $04$ has low weight. We can take $V(H_{98})=\\{12,13,23\\}$.

#### Combinatorial type $H_{99}$

The only possible multi-multiple edges are $12$, $16$, $25$, and one of $03$
or $04$. By symmetry, we can assume $04$ has low weight. We can take
$V(H_{99})=\\{01,02,05,06,12,13,16,25\\}$.

#### Combinatorial type $H_{100}$

The only possible multi-multiple edges are $03$, $04$, $13$, and $34$. We can
take $V(H_{100})=\\{01,04,12,13,14,24\\}$.

#### Combinatorial type $H_{101}$

The only possible multi-multiple edges are one of $03$ or $04$, one of $15$ or
$16$, and one of $27$ or $28$. By symmetry, we can assume $04$, $16$, and $28$
have low weight. We can take $V(H_{101})=\\{01,02,05,12,13,23\\}$.

#### Combinatorial type $H_{102}$

Note that there cannot be multi-multiple edges incident to any of $5$, $6$,
$7$, or $8$ since these are all contained in a Lannér diagram of size $4$ but
no Lánner diagram of size $3$. Moreover, no two vertices joined by a dashed
edge can be incident to a multi-multiple edge, lest the Lannér diagram of size
$2$ containing these vertices cannot be connected to the Lannér diagram
$5678$. Thus there can be at most one multi-multiple edge in any given
weighting, and by symmetry we can assume $03$ is the only multi-multiple edge.
We can then take $V(H_{102})=\\{01,03,04,12,13,23\\}$.

#### Combinatorial type $H_{103}$

The only possible multi-multiple edges are $14$ and one of $02$, $03$, $26$,
or $36$. By symmetry between vertices $2$ and $3$, we can assume $03$, $26$,
and $36$ have low weight. We can take $V(H_{103})=\\{01,06,14,24\\}$.

#### Combinatorial type $H_{104}$

The only possible multi-multiple edges are $06$, $18$, $23$, and one of $02$,
$03$, $28$, or $38$. By symmetry, we can assume $03$, $28$, and $38$ have low
weight. We can take $V(H_{104})=\\{03,06,08,12,16,26\\}$.

#### Combinatorial type $H_{105}$

The only possible multi-multiple edges are $13$, $34$, and $46$. We can take
$V(H_{105})=\\{03,04,06,13,23\\}$.

#### Combinatorial type $H_{106}$

The only possible multi-multiple edges are $18$, $26$, $34$, and one of $03$
or $04$. By symmetry, we can assume $04$ has low weight. We can take
$V(H_{106})=\\{01,05,08,12,13,14,23\\}$.

#### Combinatorial type $H_{107}$

The only possible multi-multiple edges are $34$ and $56$. We can take
$V(H_{107})=\\{04,06,13,26\\}$.

#### Combinatorial type $H_{108}$

The only possible multi-multiple edges are one of $07$ or $08$, one of $25$ or
$26$, and one of $13$ or $14$. By symmetry, we can assume $08$, $26$, and $14$
have low weight. We can take $V(H_{108})=\\{03,05,07,12,17,27\\}$.

#### Combinatorial type $H_{109}$

First, observe that the multi-multiple edges must be contained either in the
subdiagram $678$ or in the subdiagram $012345$. Moreover, there can be at most
one multi-multiple edge among $678$, lest the Lannér diagrams of size $2$
cannot be connected to the Lannér triangle. No two vertices joined by a dashed
edge can both be incident to a multi-multiple edge for this same reason.
Moreover, no vertex in $012345$ can be adjacent to two multi-multiple edges
lest these induce a Lannér triangle or violate the previous assertion. By
symmetry, we can now assume the multi-multiple edges are limited to $02$ and
$67$.

Suppose that $67$ is multi-multiple. Then neither vertex of $67$ is connected
to any vertex of $012345$ lest these induce a Lannér triangle, so for
connectivity reasons $8$ must be joined to at least one vertex of each of
$01$, $23$ and $45$. However, vertex $8$ must also be connected to a vertex of
$67$. Thus, vertex $8$ has degree at least $4$ in some elliptic diagram, which
is not possible. Therefore, we need only consider that $02$ is multi-multiple.
In this case, we can take $V(H_{109})=\\{02,04,12,13,14,24,47,67\\}$.

#### Combinatorial type $H_{110}$

The only possible multi-multiple edges are $45$, one of $34$ or $45$, one of
$24$ or $25$, and any edges of $678$. By symmetry of $4$ and $5$, we can
assume $34$ and $24$ have low weight. Since the Lannér triangle $678$ must be
connected to the other Lannér triangles, the edges $25$ and $35$ cannot
simultaneously be multi-multiple. Hence, by symmetry we can assume $35$ has
low weight. For the same reason at most one edge of $678$ can be multi-
multiple. So we can conclude, again invoking symmetry of $678$, that the
multi-multiple edges are limited to $25$, $45$, and $67$. We can then take
$V(H_{110})=\\{01,02,05,06,12,15,16,23,24,26,35\\}$.

#### Combinatorial type $H_{111}$

The only possible multi-multiple edges are $23$, $45$, one of $02$ or $03$,
and one of $14$ or $15$. By symmetry, we can assume $03$ and $15$ have low
weight. Moreover, both $02$ and $14$ cannot simultaneously be multi-multiple
lest $78$ cannot be connected to $01$, so we can assume $14$ is also low
weight. Thus, our set of multi-multiple edges must be among $23$, $45$, and
$03$. We can then set $V(H_{111})=\\{03,04,05,07,12,34,37,47\\}$.

##  8\. Remarks on Compact Coxeter $6$-Polytopes with $10$ Facets

Throughout this section, we consider compact Coxeter $6$-polytopes with $10$
facets. Using the methods outlined in Section 3, there is a finite algorithm
yielding a superset of the combinatorial types of such polytopes. Though the
process was too computationally intense for the classification to be completed
in the current project, we were able to list the combinatorial types in the
special case where the missing faces have size only $2$ and $5$. In this case,
there are precisely two combinatorial types:

$I_{1}\text{ with missing face list }\\{01,02,13,24567,34567,89\\}$

and

$I_{2}\text{ with missing face list }\\{01,02,13,24,34,56789\\}\,.$

The methods used to analyse these two combinatorial types are similar to those
used extensively in Section 7. It is straightforward to check using an
appropriate choice of $V(I_{1})$ and $V(I_{2})$ that there are no compact
Coxeter polytopes of these combinatorial types.

First, suppose there is a polytope realising $I_{1}$, and consider its Coxeter
diagram. The only multi-multiple edges are $08$, $09$, $18$, and $19$.
Moreover, note that vertices $8$ and $9$ cannot simultaneously be incident to
a multi-multiple edge, lest the Lannér diagram $89$ cannot be connected to one
of the Lannér diagrams of size $5$. Thus, by symmetry, we can assume the
multi-multiple edges are limited to $08$ and $18$. Then setting

$V(I_{1})=\\{01,03,08,09,12,18,19,28,38\\}$

is sufficient to check computationally that there are no polytopes of type
$I_{1}$.

Now consider a Coxeter diagram of combinatorial type $I_{2}$. The only
possible multi-multiple edges must be contained within the subdiagram $01234$.
Moreover, any two vertices joined by a dashed edge in this subdiagram cannot
both be incident to multi-multiple edges, lest the Lannér subdiagram
corresponding to these vertices cannot be connected to the Lannér diagram of
size $5$. Hence, there is at most one multi-multiple edge contained within
$01234$, and by symmetry we can assume this edge is $03$. One can then show
that there are no compact Coxeter polytopes of this combinatorial type by
applying Corollary 5.11 with

$V(I_{2})=\\{01,02,03,04,12,13,14,23,34\\}\,.$

The analysis of these combinatorial types as described above yields the
following result.

######  Theorem 8.1.

There are no compact Coxeter $6$-polytopes with $10$ facets having missing
faces of orders only $2$ and $5$.

There is one known compact Coxeter $6$-polytope with $10$ facets (see Figure
5), constructed by Bugaenko [7].

Figure 5. The Coxeter diagram of the compact Coxeter $6$-polytope with $10$
facets constructed by Bugaenko.

##  9\. Compact $3$-Free Coxeter Polytopes

In this section, we use a modification of Vinberg’s proof that simple Coxeter
polytopes have dimension at most $29$ to find a tighter bound on the dimension
of compact Coxeter $3$-free polytopes, i.e., polytopes where every missing
face is of size $2$. In particular, we show that the dimension of compact
$3$-free polytopes is at most $13$. While this result may be of interest in
its own right, we make use of this result to bound the dimension of polytopes
with few facets in Section 10. To our knowledge, the best previous bound on
the dimension for this class of polytopes was Vinberg’s bound $d\leq 29$ for
all compact Coxeter polytopes.

Following the terminology of Esselmann [10], a polytope is called _$k$ -free_
if it has no missing face of size at least $k$.

######  Remark 9.1.

Every face of a $k$-free polytope is itself a $k$-free polytope (see [10,
Folgerung 1.7]). This holds because given a face $f$ of a polytope $P$, any
missing face of $f$ is the intersection of a (possibly larger) missing face of
$P$ with $f$.

We first state a lemma from Vinberg’s proof of the upper bound on the
dimension of a compact polytope which relies on weightings of the planar
angles. A _planar angle_ of a polytope is a pair $(A,F)$ where $A$ is a vertex
and $F$ is a two-dimensional face containing it; $(A,F)$ is said to be a
planar angle _at_ $A$ and _on_ $F$.

######  Proposition 9.2 ([29, Prop. 6.2]).

Let $P$ be a $d$-dimensional compact Coxeter polytope and $c>0$ a positive
number. We assume that the planar angles of $P$ can be endowed with weights in
such a way that:

  1. (a)

The sum $\sigma(A)$ of the weights of the planar angles at the vertex $A$ does
not exceed $cd$.

  2. (b)

The sum $\sigma(F)$ of the weights of the planar angles of any $2$-dimensional
face is not less than $5-k$, where $k$ is the number of vertices of this face.

Then $d<8c+6$.

We also require one of Vinberg’s results on the structure of quadrilateral
two-dimensional faces. We first introduce the language of star diagrams; these
were referred to as “star schemes” by Vinberg. The _diagram of a planar angle_
$(A,F)$ is the Coxeter subdiagram $S_{A}$ with two chosen “black” vertices
corresponding to the facets containing $A$ but not $F$. Note that this is an
elliptic diagram of order $d$. The _star diagram of a face_ $F$ is the Coxeter
subdiagram $S_{F}^{*}$ whose vertices correspond to the facets having a non-
trivial intersection with $F$. We call the vertices of $S_{F}^{*}$ that are
also vertices of $S_{F}$, i.e., that correspond to facets containing $F$,
“white” and the remaining vertices “black”. Note that when $F$ is a two-
dimensional face, the black vertices correspond to facets whose intersection
with $F$ is precisely an edge.

######  Proposition 9.3 ([29, Prop. 6.4]).

Let $S_{F}^{*}$ be the star diagram of a quadrilateral two-dimensional face
$F$ of a polytope $P$. We divide the black vertices of $S_{F}^{*}$ into pairs
in such a way that the vertices corresponding to opposite sides of $F$
correspond to a single pair. Then

  1. (1)

the removal from $S_{F}^{*}$ of any two black vertices in different pairs
leaves the diagram of one of the planar angles of $F$;

  2. (2)

every hyperbolic subdiagram of $S_{F}^{*}$ contains both the black vertices
from some pair;

  3. (3)

the removal from $S_{F}^{*}$ of the two black vertices in the same pair leaves
a hyperbolic diagram.

######  Theorem 9.4.

There are no compact $3$-free Coxeter hyperbolic polytopes of dimension $14$
or higher.

###### Proof.

Let $P$ be a compact $3$-free Coxeter $d$-polytope and $S$ its scheme. We
attach weights to the planar angle $P$ similarly to the original construction
by Vinberg [29, Theorem 6.1]: the weight of a planar angle $P$ is $1$ if the
black vertices in its scheme are adjacent and $0$ otherwise. We now check that
the hypotheses of Proposition 9.2 are satisfied with $c=1$.

Since every elliptic Coxeter diagram is a forest (when viewed as an unweighted
graph), there are at most $d-1$ edges, i.e., pairs of vertices of distance at
most $c=1$. Thus, Condition (a) of Proposition 9.2 holds.

It remains to check Condition (b) for two-dimensional triangular and
quadrilateral faces. Faces with five or more vertices vacuously satisfy this
condition, as the planar angle weights are all non-negative.

First, we show that $P$ cannot have a two-dimensional triangular face. By
Remark 9.1, every face of $P$ is $3$-free as well. Note that a two-dimensional
triangular face itself has three one-dimensional facets, which all together
have trivial intersection but of which any two intersect non-trivially. Thus,
this triangular face, viewed as a two-dimensional polytope, has a missing face
of order $3$. Hence, it is not $3$-free, contradicting Remark 9.1.

Suppose now that $F$ is a two-dimensional quadrilateral face of $P$. It
follows from Proposition 9.3 that every pair of black vertices of $S_{F}^{*}$
is contained in some hyperbolic subdiagram not containing the black vertices
of the other pair. Since we have no missing faces of size other than $2$, this
implies that each pair of black vertices corresponds to Lannér subdiagram of
size $2$; call these subdiagrams $L$ and $M$. Since the subdiagram induced by
$L\cup M$ is connected (lest it be parabolic), there must be two black
vertices from separate pairs that are adjacent. These two black vertices lie
in the scheme of the corresponding angle of $F$ at a distance of $1$. Thus, we
have shown $\sigma(F)\geq 1$, as desired. Therefore, by Proposition 9.2, we
have $d<8(1)+6=14$. ∎

##  10\. Bounding the Dimension of Polytopes with Few Facets

We now utilise the classification of compact Coxeter $d$-polytopes with few
facets to place upper bounds on the dimensions of polytopes with $d+k$ facets
for $5\leq k\leq 10$. Our methods also utilise the bound $d\leq 14$ for
compact $3$-free Coxeter polytopes derived in Section 9.

######  Definition 10.1.

Let $D(k)$ denote the maximum positive integer for which a compact Coxeter
$D(k)$-polytope with $D(k)+k$ facets exists.

The only previously published bound on the dimension of such polytopes is the
following general result of Vinberg:

######  Theorem 10.2 ([30, Theorem 4]).

There are no compact Coxeter polytopes in dimension $30$ or higher.

That is, we have $D(k)\leq 29$ for all $k$. For $k\leq 4$, the exact value of
$D(k)$ is known.

######  Theorem 10.3.

222Anna Felikson and Pavel Tumarkin have shown in unpublished notes that
$D(5)\leq 8$ using a fairly involved argument [Tumarkin, personal
communication (2021)].

We have

  * •

$D(1)=4$, due to Lannér _[22]_;

  * •

$D(2)=5$, due to Kaplinskaja _[20]_ and Esselmann _[10]_;

  * •

$D(3)=8$, due to Esselmann _[10]_;

  * •

$D(4)=7$, due to Felikson and Tumarkin _[12]_.

Thus, we begin our investigation with $k=5$, and proceed until the bounds
obtained by our methods are no stronger than Vinberg’s Theorem 10.2. In
particular, we derive the following bounds in this section:

######  Theorem 10.4.

We have

  * •

$D(5)\leq 9$,

  * •

$D(6)\leq 12$,

  * •

$D(7)\leq 15$,

  * •

$D(8)\leq 18$,

  * •

$D(9)\leq 22$, and

  * •

$D(10)\leq 26$.

The argument proceeds by first proving a linear bound on $d$ having slope $4$
(with respect to $k$), then applying the classification of polytopes with
fewer facets to slightly improve these bounds in particular cases. The first
bound is obtained by iterating part of the argument used by Felikson and
Tumarkin to bound $D(4)$ [12].

######  Lemma 10.5.

If $P$ contains a missing face of size $\ell>2$, then

$D(k)\leq\max_{1\leq i\leq k-1}D(i)+\ell-1\,.$

###### Proof.

Let $T$ be a Lannér subdiagram of order $\ell$. Let $S_{0}$ be a subdiagram of
the type

$\begin{cases}G_{2}^{(m)}\text{ for some }m\geq 4&\text{ if }\ell=3\;;\\\
H_{3}\text{ or }B_{3}&\text{ if }\ell=4\;;\\\ H_{4}\text{ or }F_{4}&\text{ if
}\ell=5\;.\end{cases}$

Since $S_{0}$ has at least one bad neighbour (the unique vertex of
$T\operatorname{\setminus}S_{0}$), by Proposition 4.6 we have that $P(S_{0})$
is a Coxeter $(d-\ell+1)$-polytope with at most $d+k-(\ell+1)=(d-\ell+1)+k-1$
facets. Hence,

$d-\ell+1\leq\max_{1\leq i\leq k-1}D(i)\,.$

This implies the desired inequality. ∎

In order to obtain a general bound from this lemma, we make use of the
following result of Esselmann.

######  Lemma 10.6 ([10, Lemma 6.7]).

A $3$-free compact Coxeter $d$-polytope has at least $2d$ facets, and in the
case of equality, it must be a $d$-cube.

######  Theorem 10.7.

For $k\geq 5$, we have

$D(k)\leq\max\left\\{\max_{i<k}D(i)+4,\min\\{k-1,13\\}\right\\}\,.$

###### Proof.

Fix a Coxeter $d$-polytope $P$ with $d+k$ facets. If $P$ is not $3$-free, then
it contains a missing face of order $3$, $4$, or $5$. The first bound then
follows from Lemma 10.5.

Now suppose $P$ is $3$-free, so by Lemma 10.6, $P$ either has at least $2d+1$
facets or is a $d$-cube. Since compact Coxeter $d$-cubes only exist in
dimension at most $5$ [19], any $3$-free compact Coxeter $d$ polytope with
$d+k$ facets satisfies $d\leq k-1$ for $k>5$. For $k=5$, we can have $3$-free
compact polytopes in dimension $5$ or less, but this is exceeded by the first
bound $\max_{i<5}\\{D(i)+4\\}=12$. We furthermore have from Theorem 9.4 that
such polytopes arise in dimension at most $13$. Hence for $k>5$, a $3$-free
compact Coxeter $d$-polytope with $d+k$ facets satisfies
$d\leq\min\\{k-1,13\\}$, from which the second bound follows. ∎

We now proceed to improving this bound for polytopes with few facets, which
involves a more detailed argument depending on the classifications of compact
Coxeter polytopes with fewer facets.

###  10.1. Polytopes with $d+5$ facets

Theorem 10.7 implies that there are no compact Coxeter $d$-polytopes with
$d+5$ facets for $d\geq 13$. In this section, we additionally show that no
such polytopes arise in dimensions $12$, $11$, and $10$.

For the first reduction, we require some knowledge about compact Coxeter
$6$-polytopes with $10$ facets. In fact, it has been verified that any such
polytope has a missing face of size $3$ or $4$ using the exhaustive methods
detailed in the first portion of this work (see Theorem 8.1).

######  Proposition 10.8.

There is no Coxeter $d$-polytope with $d+5$ facets for $d\geq 10$.

###### Proof.

Suppose there is a compact Coxeter polytope $P$ of dimension $d\geq 10$ with
at most $d+5$ facets; denote its Coxeter diagram by $\Sigma$.

First suppose that $P$ has a missing face of order $3$, and let $S_{0}$ be a
subdiagram of type $G_{2}^{(m)}$ for $m\geq 4$ contained in this missing face.
The face $P(S_{0})$ corresponding to the subdiagram $S_{0}$ is a Coxeter
polytope of dimension $d_{1}\geq 8$ with at most $d_{1}+4$ facets. This must
be the unique $8$-polytope with $11$ facets. Note that Coxeter diagram of this
polytope, $\Sigma_{S_{0}}$, contains two subdiagrams of type $H_{4}$. These
subdiagrams are preserved in the subdiagram $\overline{S_{0}}$, call them
$T_{1}$ and $T_{2}$. Moreover, there is a vertex $v\in\Sigma$ connected to
$T_{1}$ and $T_{2}$ by simple edges, as well as distinct vertices $u_{i}$
connected to $T_{i}$ by a simple edge for $i\in\\{1,2\\}$, as this is true in
$\Sigma_{S_{0}}$ and is preserved under the operations described in
Proposition 4.6. In particular, $T_{1}$ has at least two bad neighbours in
$\Sigma$. The face corresponding to the subdiagram $T_{1}$ in $\Sigma$ is then
a compact Coxeter polytope of dimension $d_{2}\geq 6$ with at most $d_{2}+3$
facets. In particular, it must have dimension $6$ with $9$ facets or dimension
$8$ with $11$. There are four such polytopes, and in each such polytope every
subdiagram of type $H_{4}$ has two bad neighbours joined by simple edges.
Thus, $T_{2}$ must have at least $3$ bad neighbours joined by simple edges in
$\Sigma$, since there are at least two in each of $\overline{S_{0}}$ and
$\overline{T_{1}}$, with at most one bad neighbour being common to both. Thus,
the face corresponding to $T_{2}$ in $\Sigma$ must be a compact Coxeter
polytope of dimension $d_{3}\geq 6$ with at most $d_{3}+2$ facets, but no such
polytope exists by Theorem 10.3. Therefore, we can assume $P$ has no missing
faces of order $3$.

Now suppose that $P$ has a missing face of order $4$. Let $S_{0}$ be a
subdiagram of type $B_{3}$ or $H_{3}$ contained in this missing face. Then the
face corresponding to $S_{0}$ is a compact Coxeter polytope of dimension
$d_{1}\geq 7$ with at most $d_{1}+4$ facets. Hence, $P(S_{0})$ is either the
unique compact Coxeter $7$-polytope with $11$ facets, or the unique compact
Coxeter $8$-polytope with $11$ facets. If $P(S_{0})$ is the $7$-polytope, then
$\Sigma_{S_{0}}$ (and hence $\overline{S_{0}}$, by Proposition 4.6) contains a
subdiagram $S_{1}$ of type $H_{4}$ with three bad neighbours, each connected
by a simple edge. Thus, the face corresponding to $S_{1}$ in $\Sigma$ is a
compact Coxeter polytope of dimension $d_{2}\geq 6$ with at most $d_{2}+2$
facets, which does not exist. Thus, $P(S_{0})$ is not the unique compact
Coxeter $7$-polytope with $11$ facets. Now supposing $P(S_{0})$ is the unique
compact Coxeter $8$-polytope with $11$ facets, the same reasoning as that used
in the previous paragraph shows that $\Sigma$ must contain a subdiagram of
type $H_{4}$ with at least three bad neighbours. This again yields a face
which is a compact Coxeter polytope of dimension $d_{3}\geq 6$ with at most
$d_{3}+2$ facets, which does not exist. Therefore, we can assume $P$ has no
missing faces of order $4$.

Thus, we can now restrict to considering polytopes with missing faces of order
only $2$ and $5$. By Lemma 10.6, $P$ must contain a missing face of size $5$.
Let $S_{0}$ be a subdiagram of type $H_{4}$ or $F_{4}$ in this missing face.
Then $P(S_{0})$ is a compact Coxeter polytope of dimension $d_{1}\geq 6$ with
at most $d_{1}+4$ facets. Moreover, since $\Sigma_{S_{0}}=\overline{S_{0}}$ by
Proposition 4.6, then $P(S_{0})$ also contains missing faces of order only $2$
and $5$. In particular, $P(S_{0})$ is either a $6$-polytope with $10$ facets
or the unique compact Coxeter $8$-polytope with $11$ facets, as the unique
compact Coxeter $7$-polytope with $11$ facets contains missing faces of order
$4$. If $P(S_{0})$ is the unique compact Coxeter $8$-polytope with $11$
facets, then we reach a contradiction by the same reasoning as above.
Otherwise, $P(S_{0})$ must be a compact Coxeter $6$-polytope with $10$ facets
containing missing faces of order only $2$ and $5$, but no such polytopes
exist by Theorem 8.1. We can now conclude that no such polytope $P$ arises. ∎

######  Corollary 10.9.

We have $D(5)\leq 9$.

######  Lemma 10.10.

If there exists a $9$-polytope with $14$ facets, then it contains no missing
faces of size $3$ nor any multi-multiple edge.

###### Proof.

Suppose that $P$ is a compact $9$-polytope with $14$ facets, and let $\Sigma$
be its Coxeter diagram. Assume $P$ contains a missing face of size $3$ or a
multi-multiple edge. In the first case, let $\Sigma_{0}$ be a subdiagram of
type $G_{2}^{(m)}$ for $m\geq 4$ contained in the missing face of size $3$. In
the latter case, let $\Sigma_{0}$ be a subdiagram of type $G_{2}^{(m)}$ for
$m\geq 6$. Then the face corresponding to $\Sigma_{0}$ is a $7$-polytope with
at most $11$ facets. It must be the unique $7$-polytope with $11$ facets
constructed by Bugaenko. Moreover, by Proposition 4.6 we can obtain the
diagram $\overline{\Sigma_{0}}$ from $\Sigma_{S_{0}}$ by possibly replacing
some double edges by simple edges, or some dashed edges by ordinary or empty
edges. Thus, there is a diagram of type $H_{4}$ in $\overline{S_{0}}$ with at
least $3$ bad neighbours, call it $\Sigma_{1}$. Moreover, $\Sigma_{1}$ must
have precisely three bad neighbours in $\Sigma$, or else the face
corresponding to $\Sigma_{1}$ is a $5$-polytope with at most $6$ facets, which
does not exist. Then the the face corresponding to $\Sigma_{1}$ in $\Sigma$ is
a $5$-polytope with precisely $7$ facets, furthermore containing the fixed
missing face of size $3$ or the multi-multiple edge. However, no such
$5$-polytope exists. ∎

######  Theorem 10.11.

In a $9$-polytope with $14$ facets, every missing face of size $5$ contains a
subdiagram of type $H_{4}$ or $F_{4}$ with at least two bad neighbours.

###### Proof.

Suppose $P$ is such a polytope containing a missing face of size $5$. Let
$\Sigma_{0}$ be a subdiagram of type $F_{4}$ or $H_{4}$. Then the face
corresponding to $\Sigma_{0}$ is a $5$-polytope $P^{\prime}$ with at most $9$
facets. Furthermore, if $\Sigma_{0}$ does not itself have two bad neighbours,
then $P^{\prime}$ has precisely $9$ facets. By Lemma 10.10, $P^{\prime}$ has
no missing faces of size $3$. By the classification in Section 7, all
$5$-polytopes with $9$ facets not containing missing faces of order $3$
contain a subdiagram of type $H_{4}$ with at least $2$ bad neighbours. ∎

###  10.2. Polytopes with $d+6$ facets

Now that we have established $D(5)\leq 9$, we obtain from Theorem 10.7 that
$D(6)\leq 13$. Using results from the previous section, we can reduce this
bound by one dimension.

######  Lemma 10.12.

There are no $13$-polytopes with $19$ facets.

###### Proof.

Suppose such a polytope exists, call it $P$. By Lemma 10.5, $P$ must have
missing faces only of size $2$ or $5$. Moreover, by Lemma 10.6, $P$ must
contain at least one missing face of size $5$. Let $\Sigma_{0}$ be a
subdiagram of $T$ of type $H_{4}$ or $F_{4}$. Considering the face
corresponding to $\Sigma_{0}$, we obtain a Coxeter polytope $P^{\prime}$ of
dimension $9$ with at most $14$ facets, and furthermore it contains only
missing faces of size $2$ or $5$. Note that in fact $P^{\prime}$ must contain
precisely $14$ facets, since $D(4)<9$. Again applying Lemma 10.6, $P^{\prime}$
must contain at least one missing face of size $5$. Thus, by Theorem 10.11,
$P^{\prime}$ must contain a subdiagram of type $F_{4}$ or $H_{4}$ with at
least two bad neighbours. Looking at the face corresponding to this subdiagram
in $\Sigma$ yields a face of $P$ that is a Coxeter $9$-polytope with at most
$13$ facets. No such Coxeter polytopes exists, so we reach a contradiction. ∎

######  Corollary 10.13.

We have $D(6)\leq 12$.

We now formulate the following lemma, where we consider faces of increasing
codimension corresponding to subdiagrams of type $H_{4}$ or $F_{4}$ in
polytopes with missing faces of sizes either $2$ or $5$.

######  Lemma 10.14.

Let $P$ be a compact Coxeter $d$-polytope with $d+k$ facets whose missing
faces have size either $2$ or $5$, and $k<d$. Suppose every subdiagram in $P$
of the type $H_{4}$ or $F_{4}$ has precisely one bad neighbour (note that such