# Lipschitz Geometry of Pairs of Normally Embedded Hölder triangles

Lev Birbrair  Departamento de Matemática, Universidade Federal do Ceará (UFC),
Campus do Pici, Bloco 914, Cep. 60455-760. Fortaleza-Ce, Brasil and Institute
of Mathematics, Jagiellonian University, Profesora Stanisława Łojasiewicza 6,
30-348 Kraków, Poland<EMAIL_ADDRESS>and  Andrei Gabrielov
Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA
<EMAIL_ADDRESS>
###### Key words and phrases:

width

###### 2010 Mathematics Subject Classification:

51F30, 14P10, 03C64

*Research supported under CNPq 302655/2014-0 grant and by Capes-Cofecub

##  1\. Introduction

The question of bi-Lipschitz classification of semialgebraic surfaces has
become in recent years one of the central questions of Metric Geometry of
Singularities. There are two natural structures of a metric space on a
connected semialgebraic set $X\subset\mathbb{R}^{n}$. The first one is the
inner distance, the length of a minimal path in $X$ connecting two points. The
second one is the outer distance, defined as the distance in $\mathbb{R}^{n}$
between two points of $X$. A germ $X$ is called _normally embedded_ (see [2])
if its inner and outer metrics are equivalent. There are three natural
equivalence relations associated with these distances. Two sets $X$ and $Y$
are inner (resp., outer) equivalent if there is a inner (resp., outer) bi-
Lipschitz homeomorphism $h:X\to Y$. The sets $X$ and $Y$ are ambient bi-
Lipschitz equivalent if the homeomorphism $h:X\to Y$ can be extended to a bi-
Lipschitz homeomorphism $H$ of the ambient space. The ambient equivalence is
stronger than the outer equivalence, and the outer equivalence is stronger
then the inner equivalence. Finiteness theorems of Mostowski and Valette (see
[9] and [10]) show that there are finitely many ambient bi-Lipschitz
equivalence classes in any semialgebraic family of semialgebraic sets.

The paper [1] of the first author presents a complete bi-Lipschitz
classification of semialgebraic surface germs at the origin with respect to
the inner metric. It is based on a canonical partition of a surface germ into
Hölder triangles and isolated arcs. The exponents of these triangles, and the
combinatorics of the graph defined by their links, constitute a complete inner
Lipschitz invariant.

The outer Lipschitz geometry of semialgebraic surface germs is considerably
more complicated, and their outer bi-Lipschitz classification is still work in
progress. An important step towards such classification was made in [3], where
classification of the germs at the origin of $\mathbb{R}^{2}$ of semialgebraic
(or, more generally, definable in a polynomially bounded o-minimal structure)
Lipschitz functions with respect to contact Lipschitz equivalence relation was
suggested. It was based on a complete combinatorial invariant of contact
Lipschitz equivalence, called _pizza_.

Another important step was made in [7], where an “abnormal” semialgebraic
surface germ was canonically partitioned into normally embedded Hölder
triangles. Several constructions and results from [7] are used in the present
paper.

Normally embedded Hölder triangles are the simplest “building blocks” of
semialgebraic surface germs: the only Lipschitz invariant of a normally
embedded Hölder triangle is its exponent. In the present paper we consider the
next, a little bit more complicated, case of a _pair_ of normally embedded
Hölder triangles: a surface germ $X=T\cup T^{\prime}$ which is the union of
two normally embedded Hölder triangles $T$ and $T^{\prime}$. Let
$f:T\to\mathbb{R}$ and $g:T^{\prime}\to\mathbb{R}$ be the distances from the
points in one of these two triangles to the other one. The pizzas of $f$ and
$g$, being contact Lipschitz invariants of these two functions, are outer
Lipschitz invariants of $X$. The first question is whether $X$ is outer bi-
Lipschitz equivalent to the union of $T$ and the graph of the distance
function $f$. Simple examples (see Fig. 4) show that the answer may be
negative. Another natural question is whether the pizzas of $f$ and $g$ are
equivalent. The answer, in general, is again negative. We show (see Theorem
3.20) that the answers to both questions are positive if the pair
$(T,T^{\prime})$ is _elementary_ (see Definition 2.10) and satisfies boundary
conditions (5). The conditions (5) appear naturally in the paper [7], where
some standard building blocks (_clusters_) are defined in the link of a
singular surface. Any two Hölder triangles in a cluster satisfy (5). Although
a pair $X$ satisfying (5) is simpler than the general pair of normally
embedded Hölder triangles, its outer Lipschitz geometry is still rather
complicated. If one considers a pair $X=T\cup T^{\prime}$ of two normally
embedded Hölder triangles, such that $T^{\prime}$ is a graph of a Lipschitz
function defined on $T$, then $X$ automatically satisfies the condition (5). A
natural question is whether the opposite is true. Suppose that a pair $X=T\cup
T^{\prime}$ of normally embedded Hölder triangles satisfies (5). Is it true
that $X$ is outer Lipschitz equivalent to the union of $T$ and the graph of a
function $f$ defined on $T$? The answer is negative, and we present several
examples when this is not true (see Section 4). In this paper we define an
outer Lipschitz invariant of a pair of normally embedded Hölder triangles
satisfying (5), called _$\sigma\tau$ -pizza_, and conjecture that it is a
complete invariant: all pairs with the same $\sigma\tau$-pizza should be outer
bi-Lipschitz equivalent.

In Section 2 we give basic definitions and reformulate the pizza invariant in
the language of _zones_ (see Definition 2.20).

In Section 3 we establish properties of elementary pairs of Hölder triangles
and give examples of non-elementary pairs for which these properties fail. We
also discuss conditions satisfied by a surface germ $X=T\cup T^{\prime}$
equivalent to the union of a Hölder triangle $T$ and the graph of the distance
function $f$ defined on $T$.

In Section 4 the $\sigma\tau$-pizza is defined. The main result of the
section, Theorem 4.13, states that it is an outer Lipschitz invariant of a
pair of normally embedded Hölder triangles satisfying (5): the
$\sigma\tau$-pizzas of outer bi-Lipschitz equivalent pairs are combinatorially
equivalent. We conjecture that the converse of Theorem 4.13 is also true, but
the proof needs some additional work.

Some remarks about the figures. Since it is practically impossible to
adequately show outer Lipschitz geometry of a surface germ in a plot, we draw
instead its link (intersection with a small sphere centered at the singular
point) indicating higher tangency orders by smaller Euclidean distances. We
hope these plots will help to create geometric intuition.

##  2\. Preliminaries

All sets, functions and maps in this paper are germs at the origin of
$\mathbb{R}^{n}$ definable in a polynomially bounded o-minimal structure over
$\mathbb{R}$ with the field of exponents $\mathbb{F}$. The simplest (and most
important in applications) examples of such structures are real semialgebraic
and (global) subanalytic sets, with ${\mathbb{F}}={\mathbb{Q}}$.

######  Definition 2.1.

A germ $X$ at the origin inherits two metrics from the ambient space: the
_inner metric_ where the distance between two points of $X$ is the length of
the shortest path connecting them inside $X$, and the _outer metric_ with the
distance between two points of $X$ being their distance in the ambient space.
A germ $X$ is _normally embedded_ if its inner and outer metrics are
equivalent.

For a point $x\in X$ and a subset $Y\subset X$ we define the _outer distance_
$dist(x,Y)=\inf_{y\in Y}|x-y|$, and the _inner distance_ $idist(x,Y)$ as the
infimum of the lengths of paths connecting $x$ with points $y\in Y$.

A _surface germ_ is a closed germ $X$ such that $\dim_{\mathbb{R}}X=2$, and it
is pure dimensional.

######  Definition 2.2.

An _arc_ in $\mathbb{R}^{n}$ is (a germ at the origin of) a mapping
$\gamma:[0,\epsilon)\rightarrow\mathbb{R}^{n}$ such that $\gamma(0)=0$. Unless
otherwise specified, we suppose that arcs are parameterized by the distance to
the origin, i.e., $|\gamma(t)|=t$. We usually identify an arc $\gamma$ with
its image in $\mathbb{R}^{n}$. The _Valette link_ of $X$ is the set $V(X)$ of
all arcs $\gamma\subset X$.

######  Definition 2.3.

_Let $f\not\equiv 0$ be (a germ at the origin of) a function defined on an arc
$\gamma$. The _order_ of $f$ on $\gamma$ is the value
$q=ord_{\gamma}f\in\mathbb{F}$ such that $f(\gamma(t))=ct^{q}+o(t^{q})$ as
$t\to 0$, where $c\neq 0$. If $f\equiv 0$ on $\gamma$, we set
$ord_{\gamma}f=\infty$._

######  Definition 2.4.

The _tangency order_ of two arcs $\gamma$ and $\gamma^{\prime}$ is defined as
$tord(\gamma,\gamma^{\prime})=ord_{\gamma}|\gamma(t)-\gamma^{\prime}(t)|$. The
tangency order of an arc $\gamma$ and a set of arcs $Z\subset V(X)$ is defined
as $tord(\gamma,Z)=\sup_{\lambda\in Z}tord(\gamma,\lambda)$. The tangency
order of two subsets $Z$ and $Z^{\prime}$ of $V(X)$ is defined as
$tord(Z,Z^{\prime})=\sup_{\gamma\in Z}tord(\gamma,Z^{\prime})$. Similarly,
$itord(\gamma,\gamma^{\prime})$, $itord(\gamma,Z)$ and $itord(Z,Z^{\prime})$
denote the tangency orders with respect to the inner metric. If $T$ is a
Hölder triangle and $\gamma$ is an arc we are going to use the notation
$tord(\gamma,T)$ instead of $tord(\gamma,V(T))$ and $itord(\gamma,T)$ instead
of $itord(\gamma,V(T))$.

The tangency order defines a non-Archimedean metric on the set of arcs: if
$tord(\gamma,\gamma^{\prime})>tord(\gamma,\gamma^{\prime\prime})$ then
$tord(\gamma^{\prime},\gamma^{\prime\prime})=tord(\gamma,\gamma^{\prime\prime})$.

######  Remark 2.5.

The inner metric on a semialgebraic set is bi-Lipschitz equivalent to a
semialgebraic metric (so-called pancake metric, see the theorem of Kurdyka and
Orro [8] and also [2]). The inner order of tangency of two arks
$itord(\gamma_{1},\gamma_{2}))$ is also defined in [6].

######  Definition 2.6.

For $\beta\in\mathbb{F}$, $\beta\geq 1$, the _standard $\beta$-Hölder
triangle_ is (a germ at the origin of) the set

(1) $T_{\beta}=\\{(x,y)\in\mathbb{R}^{2}\mid x\geq 0,\;0\leq y\leq
x^{\beta}\\}.$

The curves $\\{x\geq 0,\;y=0\\}$ and $\\{x\geq 0,\;y=x^{\beta}\\}$ are the
_boundary arcs_ of $T_{\beta}$.

######  Definition 2.7.

A $\beta$-_Hölder triangle_ is (a germ at the origin of) a set
$T\subset\mathbb{R}^{n}$ that is inner bi-Lipschitz homeomorphic to the
standard $\beta$-Hölder triangle (1). The number $\beta=\mu(T)\in\mathbb{F}$
is called the _exponent_ of $T$. The arcs $\gamma_{1}$ and $\gamma_{2}$ of $T$
mapped to the boundary arcs of $T_{\beta}$ by an inner bi-Lipschitz
homeomorphism are the _boundary arcs_ of $T$ (notation
$T=T(\gamma_{1},\gamma_{2})$). All other arcs of $T$ are its _interior arcs_.
The set of interior arcs of $T$ is denoted by $I(T)$. An arc $\gamma\subset T$
is _generic_ if $itord(\gamma,\gamma_{1})=itord(\gamma,\gamma_{2})$. The set
of generic arcs of $T$ is denoted by $G(T)$.

######  Definition 2.8.

Let $X$ be a surface germ. An arc $\gamma\in V(X)$ is _Lipschitz non-singular_
if there exists a normally embedded Hölder triangle $T\subset X$ such that
$\gamma\in I(T)$ and $\gamma\not\subset\overline{X\setminus T}$. Otherwise,
$\gamma$ is _Lipschitz singular_. A Hölder triangle $T$ is _non-singular_ if
any arc $\gamma\in I(T)$ is Lipschitz non-singular.

######  Definition 2.9.

_For a Lipschitz function $f$ defined on a Hölder triangle $T$, let_

(2) $Q_{f}(T)=\bigcup_{\gamma\in V(T)}ord_{\gamma}f.$

_It was shown in[3] that $Q_{f}(T)$ is either a point or a closed interval in
$\mathbb{F}\cup\\{\infty\\}$._

######  Definition 2.10.

_A Hölder triangle $T$ is _elementary_ with respect to a Lipschitz function
$f$ if, for any $q\in Q_{f}(T)$ and any two arcs $\gamma$ and
$\gamma^{\prime}$ in $T$ such that $ord_{\gamma}f=ord_{\gamma^{\prime}}f=q$,
the order of $f$ is $q$ on any arc in the Hölder triangle
$T(\gamma,\gamma^{\prime})\subset T$._

######  Remark 2.11.

Examples 4.4, 4.5, 4.6 in [3] make the definition 2.10 more clear.

######  Definition 2.12.

Let $T$ be a Hölder triangle and $f$ a Lipschitz function defined on $T$. For
each arc $\gamma\subset T$, the _width_ $\mu_{T}(\gamma,f)$ of $\gamma$ with
respect to $f$ is the infimum of exponents of Hölder triangles
$T^{\prime}\subset T$ containing $\gamma$ such that $Q_{f}(T^{\prime})$ is a
point. For $q\in Q_{f}(T)$ let $\mu_{T,f}(q)$ be the set of exponents
$\mu_{T}(\gamma,f)$, where $\gamma$ is any arc in $T$ such that
$ord_{\gamma}f=q$. It was shown in [3] that, for each $q\in Q_{f}(T)$, the set
$\mu_{T,f}(q)$ is finite. This defines a multivalued _width function_
$\mu_{T,f}:Q_{f}(T)\to\mathbb{F}\cup\\{\infty\\}$. If $T$ is an elementary
Hölder triangle with respect to $f$ then the function $\mu_{T,f}$ is single
valued. When $f$ is fixed, we write $\mu_{T}(\gamma)$ and $\mu_{T}$ instead of
$\mu_{T}(\gamma,f)$ and $\mu_{T,f}$.

The _depth_ $\nu_{T}(\gamma,f)$ of an arc $\gamma$ with respect to $f$ is the
infimum of exponents of Hölder triangles $T^{\prime}\subset T$ such that
$\gamma\in G(T^{\prime})$ and $Q_{f}(T^{\prime})$ is a point. By definition,
$\nu_{T}(\gamma,f)=\infty$ when there are no such triangles $T^{\prime}$.

######  Definition 2.13.

_Let $T$ be a non-singular Hölder triangle and $f$ a Lipschitz function
defined on $T$. We say that $T$ is a _pizza slice_ associated with $f$ if it
is elementary with respect to $f$ and, unless $Q_{f}(T)$ is a point,
$\mu_{T,f}(q)=aq+b$ is an affine function on $Q_{f}(T)$. If $T$ is a pizza
slice such that $Q_{f}(T)$ is not a point, then the _supporting arc_
$\tilde{\gamma}$ of $T$ with respect to $f$ is the boundary arc of $T$ such
that $\mu_{T}(\tilde{\gamma},f)=\max_{q\in Q_{f}(T)}\mu_{T,f}(q)$._

######  Proposition 2.14.

_(See[3].)_ Let $T$ be a $\beta$-Hölder triangle which is a pizza slice
associated with a non-negative Lipschitz function $f$, such that $Q=Q_{f}(T)$
is not a point. Then $\mu_{T}\not\equiv\mathrm{const}$ and the following
holds:
_(1)_ $\beta\leq\mu_{T}(q)\leq\max(q,\beta)$ for all $q\in Q$,
_(2)_ $\mu_{T}(\gamma)=\beta$ for $\gamma\in G(T)$,
_(3)_ If $\tilde{\gamma}$ is the supporting arc of $T$ with respect to $f$,
then $\mu_{T}(\gamma)=itord(\tilde{\gamma},\gamma)$ for all arcs
$\gamma\subset T$ such that $\mu_{T}(\gamma)<\mu_{T}(\tilde{\gamma})$.

######  Definition 2.15.

_(See[3].)_ _Let $f$ be a non-negative Lipschitz function defined on an
oriented $\beta$-Hölder triangle $T$. A _pizza decomposition_ of $T$ (or just
a _pizza_ on $T$) associated with $f$ is a decomposition
$\\{T_{i}\\}_{i=1}^{p}$ of $T$ into $\beta_{i}$-Hölder triangles
$T_{i}=T(\lambda_{i-1},\lambda_{i})$ ordered according to the orientation of
$T$, such that
(1) $\lambda_{0}$ and $\lambda_{p}$ are the boundary arcs of $T$,
(2) $T_{i}\cap T_{i+1}=\lambda_{i}$ for $1\leq i<p$,
(3) $T_{i}\cap T_{j}=\\{0\\}$ when $|i-j|>1$,
(4) each Hölder triangle $T_{i}$ is a pizza slice associated with $f$.
We write $q_{i}=ord_{\lambda_{i}}f$, $Q_{i}=Q_{f}(T_{i})$,
$\mu_{i}(q)=\mu_{T_{i},f}(q)$. If $Q_{i}$ is not a point, then
$\tilde{\gamma}_{i}$ denotes the supporting arc of $T_{i}$ with respect to
$f$._

######  Definition 2.16.

_A pizza decomposition $\\{T_{i}\\}$ of $T$ associated with $f$ is _minimal_
if $T_{i-1}\cup T_{i}$ is not a pizza slice associated with $f$ for any
$i>1$._

######  Definition 2.17.

_For two non-negative Lipschitz functions $f$ on $T$ and $g$ on $T^{\prime}$,
a pizza decomposition $\\{T_{i}=T(\lambda_{i-1},\lambda_{i})\\}$ of $T$
associated with $f$ is _equivalent_ to a pizza decomposition
$\\{T^{\prime}_{i}=T(\lambda^{\prime}_{i-1},\lambda^{\prime}_{i})\\}$ of
$T^{\prime}$ associated with $g$ if there is an orientation preserving inner
bi-Lipschitz homeomorphism $h:T\to T^{\prime}$ such that
$h(\lambda_{i})=\lambda^{\prime}_{i}$,
$ord_{\lambda_{i}}f=ord_{\lambda^{\prime}_{i}}g$,
$Q_{f}(T_{i})=Q_{g}(T^{\prime}_{i})$ and
$\mu_{T_{i},f}\equiv\mu_{T^{\prime}_{i},g}$, for all $i$, and moreover,
$h(\tilde{\gamma}_{i})=\tilde{\gamma}^{\prime}_{i}$ if
$Q_{f}(T_{i})=Q_{g}(T^{\prime}_{i})$ is not a point, where
$\tilde{\gamma}_{i}$ and $\tilde{\gamma}^{\prime}_{i}$ are the supporting arcs
for $T_{i}$ and $T^{\prime}_{i}$ with respect to $f$ and $g$._

######  Definition 2.18.

_Let $T$ and $T^{\prime}$ be two $\beta$-Hölder triangles. Two Lipschitz
function germs $f:(T,0)\longrightarrow(\mathbb{R},0)$ and
$g:(T^{\prime},0)\longrightarrow(\mathbb{R},0)$ are _Lipschitz contact
equivalent_ if there exist two germs of inner bi-Lipschitz homeomorphisms
$h:(T,0)\longrightarrow(T^{\prime},0)$ and
$H:(T\times\mathbb{R},0)\longrightarrow(T^{\prime}\times\mathbb{R},0)$ such
that $H(T\times\\{0\\})=T^{\prime}\times\\{0\\}$ and the following diagram is
commutative:_

(3)
$\begin{array}[]{lllll}(T,0)&\stackrel{{\scriptstyle(id,\,f)}}{{\longrightarrow}}&(T\times\mathbb{R},0)&\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}&(T,0)\\\
\,\,\,h\,\downarrow&&\;\;\;\;\;H\,\downarrow&&\,\,\,h\,\downarrow\\\
(T^{\prime},0)&\stackrel{{\scriptstyle(id,\,g)}}{{\longrightarrow}}&(T^{\prime}\times\mathbb{R},0)&\stackrel{{\scriptstyle\pi^{\prime}}}{{\longrightarrow}}&(T^{\prime},0)\\\
\end{array}$

_Here $\pi:T\times\mathbb{R}\to T$ and
$\pi^{\prime}:T^{\prime}\times\mathbb{R}\to T^{\prime}$ are natural
projections._

The main result of [3], reformulated for non-negative Lipschitz functions
defined on Hölder triangles, is the following theorem.

######  Theorem 2.19.

Let $T$ and $T^{\prime}$ be oriented Hölder triangles. Non-negative Lipschitz
functions $f:T\to\mathbb{R}$ and $g:T^{\prime}\to\mathbb{R}$ are Lipschitz
contact equivalent if and only if a minimal pizza decomposition of $T$
associated with $f$ and a minimal pizza decomposition of $T^{\prime}$
associated with $g$ are equivalent. In particular, any two minimal pizza
decompositions associated with the same function $f:T\to\mathbb{R}$ are
equivalent.

######  Definition 2.20.

(See [7, Definition 2.34].) Let $X$ be a surface germ. A non-empty set of arcs
$Z\subset V(X)$ is called a _zone_ if, for any two arcs
$\gamma_{1}\neq\gamma_{2}$ in $Z$, there exists a non-singular Hölder triangle
$T=T(\gamma_{1},\gamma_{2})$ such that $V(T)\subset Z$. A _singular zone_ is a
zone $Z=\\{\gamma\\}$ consisting of a single arc $\gamma$. A zone $Z$ is
_normally embedded_ if, for any two arcs $\gamma_{1}\neq\gamma_{2}$ in $Z$,
there exists a normally embedded Hölder triangle $T=T(\gamma_{1},\gamma_{2})$
such that $V(T)\subset Z$.

######  Definition 2.21.

_(See[7, Definition 2.37].) The _order_ of a zone $Z$ is defined as
$\mu(Z)=\inf_{\gamma,\gamma^{\prime}\in Z}tord(\gamma,\gamma^{\prime})$. If
$Z$ is a singular zone then $\mu(Z)=\infty$. If $\mu(Z)=\beta$ then $Z$ is
called a $\beta$-zone._

######  Definition 2.22.

_(See[7, Definition 2.40].) A $\beta$-zone $Z$ is _closed_ if there are two
arcs $\gamma$ and $\gamma^{\prime}$ in $Z$ such that
$tord(\gamma,\gamma^{\prime})=\beta$. Otherwise, $Z$ is an _open_ zone. By
definition, any singular zone is closed._

######  Definition 2.23.

_A zone $Z\subset V(X)$ is _perfect_ if, for any two arcs $\gamma$ and
$\gamma^{\prime}$ in $Z$, there exists a Hölder triangle $T\subset X$ such
that $V(T)\subset Z$ and both $\gamma$ and $\gamma^{\prime}$ are generic arcs
of $T$. By definition, any singular zone is perfect._

######  Definition 2.24.

Let $f:T\to\mathbb{R}$ be a Lipschitz function defined on a non-singular
Hölder triangle $T$. A zone $Z\subset V(T)$ is a $q$-_order zone_ for $f$ if
$ord_{\gamma}f=q$ for any arc $\gamma\in Z$. A $q$-order zone for $f$ is
_maximal_ if it is not a proper subset of any other $q$-order zone for $f$.
The _width zone_ $W_{T}(\gamma,f)$ of an arc $\gamma\subset T$ with respect to
$f$ is the maximal $q$-order zone for $f$ containing $\gamma$, where
$q=ord_{\gamma}f$. The order of $W_{T}(\gamma,f)$ is $\mu_{T}(\gamma,f)$. The
_depth zone_ $D_{T}(\gamma,f)$ of an arc $\gamma\subset T$ with respect to $f$
is the union of zones $G(T^{\prime})$ for all triangles $T^{\prime}\subset T$
such that $\gamma\in G(T^{\prime})$ and $Q_{f}(T^{\prime})$ is a point. By
definition, $D_{T}(\gamma,f)=\\{\gamma\\}$ when there are no such triangles
$T^{\prime}$. The order of $D_{T}(\gamma,f)$ is $\nu_{T}(\gamma,f)$.

######  Lemma 2.25.

Let $f:T\to\mathbb{R}$ be a Lipschitz function defined on a non-singular
Hölder triangle $T$. For any arc $\gamma\subset T$, the width zone
$W_{T}(\gamma,f)$ is closed.

###### Proof.

If $f|_{\gamma}\equiv 0$, then either $\gamma$ is an isolated arc in the
closed subset $T_{0}=\\{f(x)=0\\}$ of $T$ and a singular zone
$W_{T}(\gamma,f)=\\{\gamma\\}$ is closed by definition, or there is a maximal
Hölder triangle $\tilde{T}_{0}\subset T_{0}$ containing $\gamma$. Then
$\mu=\mu_{T}(\gamma,f)$ is the exponent of $\tilde{T}_{0}$, and
$W_{T}(\gamma,f)=V(\tilde{T}_{0})$ is a closed $\mu$-zone. Otherwise, let
$f(\gamma(t))=c_{0}t^{q}+o(t^{q})$ where $c_{0}\neq 0$, and let
$\tilde{T}_{c}$ be the maximal Hölder triangle containing $\gamma$ in the
subset $T_{c}=\\{|f(x)|\leq ct^{q}\\}$ of $T$, where $c\geq|c_{0}|$. Then the
family $\\{\tilde{T}_{c}\\}$ is definable, Hölder triangles $\tilde{T}_{c}$
have the same exponent $\mu=\mu_{T}(\gamma,f)$ for large enough $c$, and
$W_{T}(\gamma,f)=\bigcup_{c\geq|c_{0}|}V(\tilde{T}_{c})$. Thus
$W_{T}(\gamma,f)$ is a closed $\mu$-zone. ∎

######  Definition 2.26.

_Let $T$ be a non-singular Hölder triangle and $f$ a Lipschitz function
defined on $T$. If $Z\subset V(T)$ is a zone, we define $Q_{f}(Z)$ as the set
of all exponents $ord_{\gamma}f$ for $\gamma\in Z$. The zone $Z$ is
_elementary_ with respect to $f$ if the set of arcs $\gamma\in Z$ such that
$ord_{\gamma}f=q$ is a zone for each $q\in Q_{f}(Z)$._

_For $\gamma\in Z$ and $q=ord_{\gamma}f$, the _width_ $\mu_{Z}(\gamma,f)$ of
$\gamma$ with respect to $f$ is the infimum of exponents of Hölder triangles
$T^{\prime}$ containing $\gamma$ such that $V(T^{\prime})\subset Z$ and
$Q_{f}(T^{\prime})$ is a point. The _width zone_ $W_{Z}(\gamma,f)$ of $\gamma$
with respect to $f$ is the maximal subzone of $Z$ containing $\gamma$ such
that $q=ord_{\lambda}f$ for all arcs $\lambda\subset W_{Z}(\gamma,f)$. The
order of $W_{Z}(\gamma,f)$ is $\mu_{Z}(\gamma,f)$. For $q\in Q_{f}(Z)$ let
$\mu_{Z,f}(q)$ be the set of exponents $\mu_{Z}(\gamma,f)$, where $\gamma\in
Z$ is any arc such that $ord_{\gamma}f=q$. It follows from [3] that, for each
$q\in Q_{f}(Z)$, the set $\mu_{Z,f}(q)$ is finite. This defines a multivalued
_width function_ $\mu_{Z,f}:Q_{f}(Z)\to\mathbb{F}\cup\\{\infty\\}$. If $Z$ is
an elementary zone with respect to $f$ then the function $\mu_{Z,f}$ is single
valued._

_We say that $Z$ is a _pizza slice zone_ associated with $f$ if it is
elementary with respect to $f$, $Q_{f}(Z)$ is a closed interval in
$\mathbb{F}\cup\\{\infty\\}$ and, unless $Q_{f}(Z)$ is a point,
$\mu_{Z,f}(q)=aq+b$ is an affine function on $Q_{f}(Z)$. If $Z$ is a pizza
slice zone such that $Q_{f}(Z)$ is not a point, then the _supporting subzone_
$\tilde{Z}$ of $Z$ with respect to $f$ is the set of arcs $\lambda\in Z$ such
that $\mu_{Z}(\lambda,f)=\max_{q\in Q_{f}(Z)}\mu_{Z,f}(q)$._

######  Lemma 2.27.

Let $f$ be a Lipschitz function defined on a non-singular Hölder triangle $T$.
Let $\gamma$ be an interior arc of $T$, so that $T=T^{\prime}\cup
T^{\prime\prime}$ and $T^{\prime}\cap T^{\prime\prime}=\\{\gamma\\}$. Then
either $\mu_{T^{\prime}}(\gamma,f)=\mu_{T^{\prime\prime}}(\gamma,f)$ and
$\nu_{T}(\gamma,f)=\mu_{T}(\gamma,f)$, or
$\nu_{T}(\gamma,f)=\max(\mu_{T^{\prime}}(\gamma,f),\mu_{T^{\prime\prime}}(\gamma,f))>\mu_{T}(\gamma,f)$.
In both cases, $D_{T}(\gamma,f)$ is a closed perfect zone.

###### Proof.

Let $\mu=\mu_{T}(\gamma,f)$, $\mu^{\prime}=\mu_{T^{\prime}}(\gamma,f)$ and
$\mu^{\prime\prime}=\mu_{T^{\prime\prime}}(\gamma,f)$. By definition of the
width, $\mu=\min(\mu^{\prime},\mu^{\prime\prime})$. By definition of the
depth, $\nu_{T}(\gamma,f)\geq\max(\mu^{\prime},\mu^{\prime\prime})$. According
to Lemma 2.25, the width zones $W_{T^{\prime}}(\gamma,f)$ and
$W_{T^{\prime\prime}}(\gamma,f)$ are closed zones of orders $\mu^{\prime}$ and
$\mu^{\prime\prime}$. If $\mu^{\prime}=\mu^{\prime\prime}=\mu$ then there are
two arcs $\gamma^{\prime}\subset W_{T^{\prime}}(\gamma,f)$ and
$\gamma^{\prime\prime}\subset W_{T^{\prime\prime}}(\gamma,f)$ such that
$tord(\gamma,\gamma^{\prime})=tord(\gamma,\gamma^{\prime\prime})=\mu$ and
$ord_{\lambda}f=ord_{\gamma}f$ for all arcs $\lambda\subset
T(\gamma^{\prime},\gamma^{\prime\prime})$. Then $\gamma$ is a generic arc of a
$\mu$-Hölder triangle $T(\gamma^{\prime},\gamma^{\prime\prime})$, thus
$\nu_{T}(\gamma,f)\leq\mu$. Since $\nu_{T}(\gamma,f)\geq\mu$, we have
$\nu_{T}(\gamma,f)=\mu$ in this case. Otherwise, if
$\mu^{\prime}>\mu^{\prime\prime}$ then, according to Lemma 2.25, there are two
arcs $\gamma^{\prime}\subset W_{T^{\prime}}(\gamma,f)$ and
$\gamma^{\prime\prime}\subset W_{T^{\prime\prime}}(\gamma,f)$ such that
$tord(\gamma,\gamma^{\prime})=tord(\gamma,\gamma^{\prime\prime})=\mu^{\prime}$
and $ord_{\lambda}f=ord_{\gamma}f$ for all arcs $\lambda\subset
T(\gamma^{\prime},\gamma^{\prime\prime})$. Then $\gamma$ is a generic arc of a
$\mu^{\prime}$-Hölder triangle $T(\gamma^{\prime},\gamma^{\prime\prime})$,
thus $\nu_{T}(\gamma,f)\leq\mu^{\prime}$. Since
$\nu_{T}(\gamma,f)\geq\max(\mu^{\prime},\mu^{\prime\prime})$, we have
$\nu_{T}(\gamma,f)=\max(\mu^{\prime},\mu^{\prime\prime})$ in this case.

To show that $D_{T}(\gamma,f)$ is a closed perfect zone, note first that its
order is $\nu=\nu_{T}(\gamma,f)$ and, unless $\nu=\infty$ and
$D_{T}(\gamma,f)=\\{\gamma\\}$ is by definition closed perfect, $\gamma$ is a
generic arc of a $\nu$-Hölder triangle
$\tilde{T}=T(\gamma^{\prime},\gamma^{\prime\prime})\subset T$ such that
$tord_{\lambda}f=tord_{\gamma}f$ for any arc $\lambda\subset\tilde{T}$. Then,
since
$tord(\gamma,\gamma^{\prime})=tord(\gamma,\gamma^{\prime\prime})=\nu<\infty$,
there is a generic arc $\lambda$ of $\tilde{T}$ such that
$tord(\lambda,\gamma)=\nu$, thus $\bar{T}=T(\lambda,\gamma)$ is a $\nu$-Hölder
triangle and $V(\bar{T})\subset D_{T}(\gamma,f)$. This implies that
$D_{T}(\gamma,f)$ is a closed zone. If $\lambda^{\prime}$ and
$\lambda^{\prime\prime}$ are any two arcs in $D_{T}(\gamma,f)$, then there are
two Hölder triangles $T^{\prime}\subset T$ and $T^{\prime\prime}\subset T$
containing $\gamma$ such that $\lambda^{\prime}\in G(T^{\prime})$ and
$\lambda^{\prime\prime}\in G(T^{\prime\prime})$. Then both $\lambda^{\prime}$
and $\lambda^{\prime\prime}$ are generic arcs of $T^{\prime}\cup
T^{\prime\prime}$, thus $D_{T}(\gamma,f)$ is a perfect zone. ∎

######  Remark 2.28.

_Let $h:T\to T^{\prime}$ be an inner bi-Lipschitz homeomorphism, and let
$f(x)=g(h(x))$ where $g$ is a Lipschitz function defined on $T^{\prime}$. Then
$\mu_{T}(\gamma,f)=\mu_{T^{\prime}}(h(\gamma),g)$,
$\nu_{T}(\gamma,f)=\nu_{T^{\prime}}(h(\gamma),g)$,
$h(W_{T}(\gamma,f))=W_{T^{\prime}}(h(\gamma),g)$ and
$h(D_{T}(\gamma,f))=D_{T^{\prime}}(h(\gamma),g)$, for any arc $\gamma\in
V(T)$._

######  Lemma 2.29.

A zone $Z\subset V(X)$ is perfect if and only if, for any two arcs $\gamma$
and $\gamma^{\prime}$ in $Z$, there exists a Hölder triangle $T\subset X$ such
that $V(T)\subset Z$, and an inner bi-Lipschitz automorphism $h:X\to X$ such
that $h(\gamma)=\gamma^{\prime}$ and $h(x)=x$ for all $x\in X\setminus T$.

###### Proof.

Let $Z\subset V(X)$ be a perfect zone and $\gamma,\gamma^{\prime}$ two arcs in
$Z$. Let $T=T(\gamma_{1},\gamma_{2})$ be a $\beta$-Hölder triangle such that
$V(T)\subset Z$ and both $\gamma$ and $\gamma^{\prime}$ are generic arcs in
$T$. Then $T=T(\gamma_{1},\gamma)\cup T(\gamma,\gamma_{2})$ and
$T=T(\gamma_{1},\gamma^{\prime})\cup T(\gamma^{\prime},\gamma_{2})$ are two
decompositions of $T$ into $\beta$-Hölder triangles. Let
$h_{1}:T(\gamma_{1},\gamma)\to T(\gamma_{1},\gamma^{\prime})$ and
$h_{2}:T(\gamma,\gamma_{2})\to T(\gamma^{\prime},\gamma_{2})$ be inner bi-
Lipschitz homeomorphisms, such that $h_{1}|_{\gamma_{1}=Id}$,
$h_{2}|_{\gamma_{2}}=Id$ and $h_{1}|_{\gamma}=h_{2}|_{\gamma}$. Then the
mapping $h:T\to T$ such that $h=h_{1}$ on $T(\gamma_{1},\gamma)$ and $h=h_{2}$
on $T(\gamma,\gamma_{2})$ is an inner bi-Lipschitz homeomorphism such that
$h|_{\gamma_{1}}=Id$, $h|_{\gamma_{2}}=Id$ and $h(\gamma)=\gamma^{\prime})$.
Thus $h$ can be extended by identity outside $T$ to an inner bi-Lipschitz
homeomorphism $X\to X$ preserving $Z$. ∎

######  Proposition 2.30.

Let $f$ be a non-negative Lipschitz function defined on a normally embedded
Hölder triangle $T=T(\gamma_{1},\gamma_{2})$, oriented from $\gamma_{1}$ to
$\gamma_{2}$. There exists a unique finite family
$\\{D_{\ell}\\}_{\ell=0}^{p}$ of disjoint zones $D_{\ell}\subset V(T)$, the
pizza zones associated with $f$, with the following properties:
_1._ The singular zones $D_{0}=\\{\gamma_{1}\\}$ and $D_{p}=\\{\gamma_{2}\\}$
are the boundary arcs of $T$.
_2._ For any arc $\gamma\in D_{\ell}$, $D_{\ell}=D_{T}(\gamma,f)$ is a closed
perfect $\nu_{\ell}$-zone, where $\nu_{\ell}=\nu_{T}(\gamma,f)$. In
particular, $D_{\ell}$ is a $q_{\ell}$-order zone for $f$, where
$q_{\ell}=ord_{\gamma}f$ for $\gamma\in D_{\ell}$. Moreover, $D_{\ell}$ is a
maximal $q_{\ell}$-order zone for $f$ of order $\nu_{\ell}$: if $Z\subset
V(T)$ is a $q_{\ell}$-order zone for $f$ containing $D_{\ell}$ and $\lambda\in
Z$ is an arc such that $tord(\lambda,D_{\ell})\geq\nu_{\ell}$, then
$\lambda\in D_{\ell}$.
_3._ Any choice of arcs $\lambda_{\ell}\in D_{\ell}$ defines a minimal pizza
$\\{T_{\ell}=T(\lambda_{\ell-1},\lambda_{\ell})\\}_{\ell=1}^{p}$ on $T$
associated with $f$.
_4._ Any minimal pizza on $T$ associated with $f$ can be obtained as a
decomposition $\\{T_{\ell}\\}$ of $T$ defined by some choice of arcs
$\lambda_{\ell}\in D_{\ell}$.

###### Proof.

Consider a decomposition $\\{T_{\ell}\\}_{\ell=1}^{p}$, of $T$ into
$\beta_{\ell}$-Hölder triangles $T_{\ell}=T(\lambda_{\ell-1},\lambda_{\ell})$
which is a minimal pizza for $f$. Let
$Q_{\ell}\subset\mathbb{F}\cup\\{\infty\\}$ be the set (either a point or a
closed interval) of values $tord_{\gamma}f$ for $\gamma\subset T_{\ell}$, and
let $\mu_{\ell}:Q_{\ell}\to\mathbb{F}\cup\\{\infty\\}$ be the affine width
function for $f$ on $T_{\ell}$ (a constant if $Q_{\ell}$ is a point). We
assume that $\lambda_{0}$ and $\lambda_{p}$ are the boundary arcs of $T$, and
that $T_{\ell}\cap T_{\ell+1}=\lambda_{\ell}$ for $1\leq\ell<p$.

Since each boundary arc of $T$ is also a boundary arc of a pizza slice for any
pizza decomposition of $T$, we can define singular zones
$D_{0}=\\{\lambda_{0}\\}$ and $D_{p}=\\{\lambda_{p}\\}$.

If the germ at zero of the set $S=\\{x\in T,\;f(x)=0\\}$ is non-empty, it is a
union of finitely many germs isolated arcs and germs of maximal in $S$ Hölder
triangles $S_{j}$. Each isolated arc of $S$, and each boundary arc of one of
the triangles $S_{j}$, must be a boundary arc of a pizza slice for any minimal
pizza on $T$ associated with $f$. In particular, such an arc $\lambda$ must be
one of the arcs $\lambda_{\ell}$, and the singular zone $\\{\lambda\\}$ must
be one of the zones $D_{\ell}$.

Assume now that $0<\ell<p$ and $q_{\ell}=ord_{\lambda_{\ell}}f<\infty$.
Consider the depth zone $D_{\ell}=D_{T}(\lambda_{\ell},f)$ (see Definition
2.24). Then $D_{\ell}$ is a closed perfect zone of order
$\nu_{\ell}=\nu_{T}(\lambda_{\ell},f)$, which is also a $q_{\ell}$-order zone
for $f$. Moreover, if $\lambda\subset T_{\ell}$ is an arc such that
$tord(\lambda,\lambda_{\ell})\geq\nu_{\ell}$ and $ord_{\gamma}f=q_{\ell}$ for
any arc $\gamma\subset T(\lambda_{\ell},\lambda)$, then $\lambda\in D_{\ell}$
by Definition 2.24. The same argument works for $\lambda\subset T_{\ell-1}$.
Thus $D_{\ell}$ is a maximal $q_{\ell}$-order zone for $f$ of order
$\nu_{\ell}$.

We claim that, if the arc $\lambda_{\ell}$ is replaced by any other arc
$\theta\in D_{\ell}$ and the Hölder triangles
$T_{\ell}=T(\lambda_{\ell-1},\lambda_{\ell})$ and
$T_{\ell+1}=T(\lambda_{\ell},\lambda_{\ell+1})$ with the common arc
$\lambda_{\ell}$ are replaced by the Hölder triangles
$T(\lambda_{\ell-1},\theta)$ and $T(\theta,\lambda_{\ell+1})$ with the common
arc $\theta$, the resulting decomposition of $T$ is again a minimal pizza on
$T$ associated with $f$. Indeed, since $D_{\ell}$ is a perfect zone, and also
a $q_{\ell}$-order zone for $f$, by Lemma 2.29 one can construct an inner bi-
Lipschitz map $\phi:T\to T$, such that $\phi(\lambda_{\ell})=\theta$ and
$\phi(\gamma)=\gamma$ for any arc $\gamma\in V(T)\setminus D_{\ell}$. In
particular, $ord_{\phi(\gamma)}f=ord_{\gamma}f$ for each arc $\gamma\subset
T$, thus $\phi$ transforms the function $f$ into a $v$-equivalent function.
This implies that $\phi$ preserves all zones $D_{\ell}$, and that
decomposition $\\{\phi(T_{\ell})\\}$ defines a minimal pizza on $T$ associated
with $f$. Replacing all arcs $\lambda_{\ell}$ with some other arcs
$\theta_{\ell}\in D_{\ell}$, for $\ell=0,\dots,p$, we see that any choice of
arcs $\lambda_{\ell}\in D_{\ell}$ results in a minimal pizza on $T$ associated
with $f$.

On the other hand, given a minimal pizza
$\\{T_{\ell}=T(\lambda_{\ell-1},\lambda_{\ell})\\}$ on $T$ associated with
$f$, consider any other minimal pizza
$\\{T^{\prime}_{\ell}=T(\theta_{\ell-1},\theta_{\ell})\\}$ on $T$ associated
with $f$. By the Lipschitz contact invariance of a minimal pizza (see Theorem
2.19) there exists an inner bi-Lipschitz homeomorphism $h:T\to T$ such that
$h(\lambda_{\ell})=\theta_{\ell}$ and $h(T_{\ell})=T^{\prime}_{\ell}$ for all
$\ell$, and also such that $ord_{h(\gamma)}f=ord_{\gamma}f$ for any arc
$\gamma\subset T$. Thus $h$ transforms $f$ into a function of the same contact
(see Definition 2.2 from [4]) . Since the zones $D_{\ell}$ are Lipschitz
invariant, we have $h(D_{\ell})=D_{\ell}$ for all $\ell$, thus
$\theta_{\ell}\in D_{\ell}$. This proves that any minimal pizza can be
obtained by some choice of arcs $\lambda_{\ell}\in D_{\ell}$. ∎

######  Corollary 2.31.

Let $\\{D_{\ell}\\}_{\ell=0}^{p}$ be the pizza zones of a minimal pizza
$\\{T_{\ell}=T(\lambda_{\ell-1},\lambda_{\ell})\\}_{\ell=1}^{p}$ on $T$
associated with $f$, as in Proposition 2.30. For each $\ell=1,\ldots,p$, the
set $Y_{\ell}=D_{\ell-1}\cup D_{\ell}\cup V(T_{\ell})$ is a pizza slice zone
associated with $f$, independent of the choice of arcs $\lambda_{\ell}\in
D_{\ell}$. Moreover, $Y_{\ell}$ is a maximal pizza slice zone: if a pizza
slice zone $Y\subset V(T)$ associated with $f$ contains $Y_{\ell}$ then
$Y=Y_{\ell}$.

##  3\. Elementary pairs of normally embedded Hölder triangles

Let $T=T(\gamma_{1},\gamma_{2})$ and
$T^{\prime}=T(\gamma^{\prime}_{1},\gamma^{\prime}_{2})$ be two normally
embedded $\beta$-Hölder triangles, oriented from $\gamma_{1}$ to $\gamma_{2}$
and from $\gamma^{\prime}_{1}$ to $\gamma^{\prime}_{2}$ respectively.

######  Definition 3.1.

_A pair $(\gamma,\gamma^{\prime})$ of arcs $\gamma\subset T$ and
$\gamma^{\prime}\subset T^{\prime}$ is _regular_ if_

(4)
$tord(\gamma,T^{\prime})=tord(\gamma,\gamma^{\prime})=tord(\gamma^{\prime},T).$

######  Proposition 3.2.

Let $T=T(\gamma_{1},\gamma_{2})$ and
$T^{\prime}=T(\gamma^{\prime}_{1},\gamma^{\prime}_{2})$ be two normally
embedded $\beta$-Hölder triangles. Let $f(x)=dist(x,T^{\prime})$ be the
distance from $x\in T$ to $T^{\prime}$, and let
$g(x^{\prime})=dist(x^{\prime},T)$ be the distance from $x^{\prime}\in
T^{\prime}$ to $T$. Let $\Gamma\subset T\times\mathbb{R}$ and
$\Gamma^{\prime}\subset T^{\prime}\times\mathbb{R}$ be the graphs of the
functions $f(x)$ and $g(x^{\prime})$. Then the following conditions are
equivalent:

1\. There is a homeomorphism $H:T\cup T^{\prime}\to T\cup\Gamma$, bi-Lipschitz
with respect to the outer metric, such that $H(\gamma_{1})=\gamma_{1}$ and
$H(\gamma_{2})=\gamma_{2}$.

2\. There is a homeomorphism $H^{\prime}:T\cup T^{\prime}\to
T^{\prime}\cup\Gamma^{\prime}$, bi-Lipschitz with respect to the outer metric,
such that $H^{\prime}(\gamma^{\prime}_{1})=\gamma^{\prime}_{1}$ and
$H^{\prime}(\gamma^{\prime}_{2})=\gamma^{\prime}_{2}$.

3\. There exists a bi-Lipschitz homeomorphism $h:T\to T^{\prime}$ such that
$h(\gamma_{1})=\gamma^{\prime}_{1}$, $h(\gamma_{2})=\gamma^{\prime}_{2}$ and
$tord(\gamma,h(\gamma))=tord(\gamma,T^{\prime})$ for any arc $\gamma\subset
T$.

4\. There exists a bi-Lipschitz homeomorphism $h^{\prime}:T^{\prime}\to T$
such that $h^{\prime}(\gamma^{\prime}_{1})=\gamma_{1}$,
$h^{\prime}(\gamma^{\prime}_{2})=\gamma_{2}$ and
$tord(\gamma^{\prime},h^{\prime}(\gamma^{\prime}))=tord(\gamma^{\prime},T)$
for any arc $\gamma^{\prime}\subset T^{\prime}$.

5\. There exists a bi-Lipschitz homeomorphism $h:T\to T^{\prime}$ such that
$h(\gamma_{1})=\gamma^{\prime}_{1}$, $h(\gamma_{2})=\gamma^{\prime}_{2}$, and
the pair of arcs $(\gamma,h(\gamma))$ is regular for any arc $\gamma\subset
T$.

###### Proof.

If condition 1 is satisfied, we may assume that $H(T)=T$ and
$H(T^{\prime})=\Gamma$. Since $f$ is a Lipschitz function on $T$ and $H$ is an
outer bi-Lipschitz homeomorphism, we have
$tord(\gamma,T^{\prime})=tord(H(\gamma),\Gamma)=ord_{H(\gamma)}f=tord(H(\gamma),f(H(\gamma))$
for any arc $\gamma\subset T$. Since $H^{-1}$ is also an outer bi-Lipschitz
homeomorphism, the mapping $h:T\to T^{\prime}$ defined as
$h(x)=H^{-1}((H(x),f(H(x)))$ is a bi-Lipschitz homeomorphism satisfying
condition 5, which implies conditions 3 and 4. Conversely, given a
homeomorphism $h:T\to T^{\prime}$ satisfying condition 3, the mapping $H:T\cup
T^{\prime}\to T\cup\Gamma$ which is the identity on $T$ and defined as
$H(x^{\prime})=(h^{-1}{x^{\prime}},f(h^{-1}(x^{\prime})))$for $x^{\prime}\in
T^{\prime}$ satisfies condition 1. Thus conditions 1, 3 and 5 are equivalent.

Similarly, conditions 2, 4 and 5 are equivalent.

If conditions 1 and 3 are satisfied, we may assume that $T^{\prime}=\Gamma$
and $h(x)=(x,f(x))$ for $x\in T$. Then
$tord(\gamma,T^{\prime})=tord(\gamma^{\prime},T)$ for any arcs $\gamma\subset
T$ and $\gamma^{\prime}=\\{(x,f(x)):x\in\gamma\\}\subset T^{\prime}$. Thus
$h^{\prime}=h^{-1}:T^{\prime}\to T$ satisfies condition 2. This implies that
all five conditions are equivalent. ∎

If conditions of Proposition 3.2 are satisfied then the pairs of arcs
$(\gamma_{1},\gamma^{\prime}_{1})$ and $(\gamma_{2},\gamma^{\prime}_{2})$ are
regular:

(5)
$tord(\gamma_{1},T^{\prime})=tord(\gamma_{1},\gamma^{\prime}_{1})=tord(\gamma^{\prime}_{1},T),\quad
tord(\gamma_{2},T^{\prime})=tord(\gamma_{2},\gamma^{\prime}_{2})=tord(\gamma^{\prime}_{2},T).$

In general, the opposite does not hold. However, Theorem 3.20 below states
that conditions of Proposition 3.2 are satisfied if $T$ is elementary with
respect to $f$ and (5) holds. The following Proposition from [7] is an
important step in the proof of Theorem 3.20.

######  Proposition 3.3.

_(see[7, Proposition 2.20])_ Let $T=T(\gamma_{1},\gamma_{2})$ and
$T^{\prime}=T(\gamma^{\prime}_{1},\gamma^{\prime}_{2})$ be normally embedded
$\beta$-Hölder triangles such that
$tord(\gamma_{1},\gamma^{\prime}_{1})\geq\alpha,\;tord(\gamma_{2},\gamma^{\prime}_{2})\geq\alpha$,
and $tord(\gamma,T^{\prime})\geq\alpha$ for all arcs $\gamma\subset T$, for
some $\alpha>\beta$. Then there is a bi-Lipschitz homeomorphism $h:T\to
T^{\prime}$ such that
$h(\gamma_{1})=\gamma^{\prime}_{1},\;h(\gamma_{2})=\gamma^{\prime}_{2}$, and
$tord(h(\gamma),\gamma)\geq\alpha$ for any arc $\gamma\subset T$.

######  Remark 3.4.

If a $\beta$-Hölder triangle $T=T(\gamma_{1},\gamma_{2})$ and a
$\beta^{\prime}$-Hölder triangle
$T^{\prime}=T(\gamma^{\prime}_{1},\gamma^{\prime}_{2})$ are normally embedded
and satisfy (5) then $\beta^{\prime}=\beta$, unless
$tord(T,T^{\prime})\leq\min(\beta,\beta^{\prime})$.

######  Lemma 3.5.

Let $T=T(\gamma_{1},\gamma_{2})$ and
$T^{\prime}=T(\gamma^{\prime}_{1},\gamma^{\prime}_{2})$ be normally embedded
Hölder triangles. Let $\lambda_{1}\neq\lambda_{2}$ be two arcs in $T$, and let
$\theta_{1}\subset T^{\prime},\;\theta_{2}\subset T^{\prime}$ and
$\theta\subset T(\theta_{1},\theta_{2})\subset T^{\prime}$ be three arcs such
that
$tord(\theta,T)=q<\min(tord(\lambda_{1},\theta_{1}),tord(\lambda_{2},\theta_{2}))$.
Then there is an arc $\lambda\subset T(\lambda_{1},\lambda_{2})\subset T$ such
that $tord(\lambda,T^{\prime})\leq q$.

###### Proof.

We may assume that $\theta_{1}\subset
T^{\prime}_{1}=T(\gamma^{\prime}_{1},\theta)$ and $\theta_{2}\subset
T^{\prime}_{2}=T(\theta,\gamma^{\prime}_{2})$. For $x\in T$, let
$f_{1}(x)=dist(x,T^{\prime}_{1})$ and $f_{2}(x)=dist(x,T^{\prime}_{2})$. Then
$f(x)=dist(x,T^{\prime})=\min(f_{1}(x),f_{2}(x))$. Since
$tord(\theta,\lambda_{1})\leq tord(\theta,T)=q$ and
$tord(\lambda_{1},\theta_{1})>q$, we have by the non-Archimedean property
$tord(\theta_{1},\theta)=\min(tord(\lambda_{1},\theta_{1}),tord(\lambda_{1},\theta))\leq
q$. Since $T^{\prime}$ is normally embedded, we have
$tord(\theta_{1},T^{\prime}_{2})=tord(\theta_{1},\theta)\leq q$. Since
$tord(\lambda_{1},\theta_{1})>q$, this implies
$tord(\lambda_{1},T^{\prime}_{2})=\min(tord(\lambda_{1},\theta_{1}),tord(\theta_{1},T^{\prime}_{2}))\leq
q$ by the non-Archimedean property. Thus
$f|_{\lambda_{1}}=f_{1}|_{\lambda_{1}}$. Similarly,
$f|_{\lambda_{2}}=f_{2}|_{\lambda_{2}}$, thus there is an arc $\lambda\subset
T(\lambda_{1},\lambda_{2})$ such that
$f|_{\lambda}=f_{1}|_{\lambda}=f_{2}|_{\lambda}$ (see Fig. 1). Then
$tord(\lambda,T^{\prime})=ord_{\lambda}f\leq q$, otherwise we would have
$tord(\lambda,T^{\prime}_{1})=tord(\lambda,T^{\prime}_{2})>q$. Since
$tord(\lambda,\theta)\leq tord(\theta,T)=q$, this would contradict to
$T^{\prime}=T^{\prime}_{1}\cup T^{\prime}_{2}$ being normally embedded. ∎

######  Corollary 3.6.

Let $T$ and $T^{\prime}$ be normally embedded Hölder triangles. Let
$\tilde{T}=T(\lambda_{1},\lambda_{2})\subset T$ be a $\beta$-Hölder triangle
such that $tord(\gamma,T^{\prime})=q>\beta$ for any arc
$\gamma\subset\tilde{T}$. If
$\tilde{T}^{\prime}=T(\theta_{1},\theta_{2})\subset T^{\prime}$ is a
$\beta$-Hölder triangle such that
$tord(\theta_{1},\lambda_{1})=tord(\theta_{2},\lambda_{2})=q$ then, for any
arc $\theta\subset T^{\prime}$ such that $tord(\theta,\theta_{1})<q$ and
$tord(\theta,\theta_{2})<q$, we have $tord(\theta,T)=q$.

###### Proof.

Lemma 3.5 implies that $tord(\theta,T)\geq tord(\theta,\tilde{T})\geq q$ for
any arc $\theta\subset\tilde{T}^{\prime}$. If
$\theta\subset\tilde{T}^{\prime}$ is an arc such that
$tord(\theta,\theta_{1})<q$ and $tord(\theta,\theta_{2})<q$, Proposition 3.3
implies that $tord(\theta,T\setminus\tilde{T})<q$. If $tord(\theta,T)>q$ and
$\gamma\subset T$ is an arc such that $tord(\gamma,\theta)>q$, then
$\gamma\subset\tilde{T}$ and $tord(\gamma,T^{\prime})>q$, a contradiction.
Thus $tord(\theta,T)=q$. ∎

######  Definition 3.7.

Let $T$ and $T^{\prime}$ be normally embedded oriented Hölder triangles. A
pair of $\beta$-Hölder triangles $\tilde{T}=T(\lambda_{1},\lambda_{2})\subset
T$ and $\tilde{T}^{\prime}=T(\theta_{1},\theta_{2})\subset T^{\prime}$ in
Corollary 3.6 is called _positively oriented_ if their orientations induced
from $T$ and $T^{\prime}$ are either both the same as their orientations from
$\lambda_{1}$ to $\lambda_{2}$ and from $\theta_{1}$ to $\theta_{2}$ or both
opposite. Otherwise, $\tilde{T}$ and $\tilde{T}^{\prime}$ is called a
_negatively oriented_ pair of Hölder triangles.

######  Remark 3.8.

Any pair of $\alpha$-Hölder triangles
$T(\lambda^{\prime}_{1},\lambda^{\prime}_{2})\subset\tilde{T}$ and
$T(\theta^{\prime}_{1},\theta^{\prime}_{2})\subset\tilde{T}^{\prime}$, where
$\alpha<q$, satisfying conditions
$tord(\theta^{\prime}_{1},\lambda^{\prime}_{1})=tord(\theta^{\prime}_{2},\lambda^{\prime}_{2})=q$
is positively (resp., negatively) oriented if, and only if, the pair
$(\tilde{T},\tilde{T}^{\prime})$ is positively (resp., negatively) oriented.

######  Proposition 3.9.

Let $T$ and $T^{\prime}$ be normally embedded Hölder triangles with the
distance functions $f(x)=dist(x,T^{\prime})$ and
$g(x^{\prime})=dist(x^{\prime},T)$. Let $Z\subset V(T)$ be a maximal $q$-order
zone for $f$ such that $\mu(Z)<q$. Then there exists a unique maximal
$q$-order zone $Z^{\prime}\subset V(T^{\prime})$ for $g$ such that
$\mu(Z^{\prime})=\mu(Z)$ and, for any arc $\gamma\in Z$ such that
$\nu_{Z}(\gamma,f)<q$ and any arc $\gamma^{\prime}\subset T^{\prime}$ such
that $tord(\gamma,\gamma^{\prime})=q$, we have $\gamma^{\prime}\subset
Z^{\prime}$ and $\nu_{Z^{\prime}}(\gamma^{\prime},g)=\nu_{Z}(\gamma,f)$.
Conversely, if $\gamma^{\prime}\in Z^{\prime}$ is any arc such that
$\nu_{Z^{\prime}}(\gamma^{\prime},g)<q$ then, for any arc $\gamma\subset T$
such that $tord(\gamma,\gamma^{\prime})=q$, we have $\gamma\subset Z$ and
$\nu_{Z}(\gamma,f)=\nu_{Z^{\prime}}(\gamma^{\prime},g)$.

###### Proof.

Let $\tilde{Z}\subset Z$ be the set of all arcs $\gamma\subset Z$ such that
$\nu_{Z}(\gamma,f)<q$. Let $\lambda_{1}$ and $\lambda_{2}$ be any two arcs in
$\tilde{Z}$ such that $\beta=tord(\lambda_{1},\lambda_{2})<q$. Consider the
$\beta$-Hölder triangle $\tilde{T}=T(\lambda_{1},\lambda_{2})\subset T$. Since
$Z$ is a zone, we have $V(\tilde{T})\subset Z$, thus
$tord(\gamma,T^{\prime})=q$ for any arc $\gamma\subset\tilde{T}$. Also,
$\nu_{Z}(\gamma,f)\leq\max(\nu_{Z}(\lambda_{1},f),\nu_{Z}(\lambda_{2},f))<q$
for any arc $\gamma\subset\tilde{T}$, thus $V(\tilde{T})\subset\tilde{Z}$.
This implies that $\tilde{Z}$ is a $q$-order zone for $f$. Let
$\tilde{Z}^{\prime}$ be the set of all arcs $\gamma^{\prime}\subset
T^{\prime}$ such that $tord(\gamma,\gamma^{\prime})=q$ for some arc
$\gamma\subset\tilde{Z}$.

Let $\lambda^{\prime}_{1}$ and $\lambda^{\prime}_{2}$ be any two arcs in
$T^{\prime}$ such that
$tord(\lambda_{1},\lambda^{\prime}_{1})=tord(\lambda_{2},\lambda^{\prime}_{2})=q$.
Since $\beta<q$,
$\tilde{T}^{\prime}=T(\lambda^{\prime}_{1},\lambda^{\prime}_{2})\subset
T^{\prime}$ is a $\beta$-Hölder triangle. Since $\lambda_{1}\in\tilde{Z}$ and
$\lambda_{2}\in\tilde{Z}$, we have
$V(\tilde{T}^{\prime})\subset\tilde{Z}^{\prime}$. It follows from Proposition
3.3 that $tord(\gamma,T^{\prime}\setminus\tilde{T}^{\prime})<q$ for any arc
$\gamma\subset\tilde{T}$, thus $tord(\gamma,\tilde{T}^{\prime})=q$ for any arc
$\gamma\subset\tilde{T}$. Corollary 3.6 implies that
$tord(\theta,\tilde{T})=q$ for any arc $\theta\subset\tilde{T}^{\prime}$. This
implies that $\tilde{Z}^{\prime}$ is a $q$-order zone for $g$. It follows from
the non-Archimedean property that
$\nu_{\tilde{Z}^{\prime}}(\gamma^{\prime},g)<q$ for any arc
$\gamma^{\prime}\in\tilde{Z}^{\prime}$.

Let $Z^{\prime}$ be the maximal $q$-zone for $g$ containing
$\tilde{Z}^{\prime}$. By the construction this zone is unique. Let us show
that $\nu_{Z^{\prime}}(\gamma^{\prime},g)\geq q$ for any arc
$\gamma^{\prime}\in Z^{\prime}\setminus\tilde{Z}^{\prime}$. If
$\gamma^{\prime}\in Z^{\prime}\setminus\tilde{Z}^{\prime}$ and
$\nu_{Z^{\prime}}(\gamma^{\prime},g)<q$, applying the same arguments as above
to $Z^{\prime}$ and $g$ instead of $Z$ and $f$, we can show that any arc
$\gamma\subset T$ such that $tord(\gamma,\gamma^{\prime})=q$ belongs to $Z$
and $\nu_{Z}(\gamma,f)<q$. Thus $\gamma\in\tilde{Z}$, which implies
$\gamma^{\prime}\in\tilde{Z}^{\prime}$, a contradiction. The equality
$\nu_{Z}(\gamma,f)=\nu_{Z^{\prime}}(\gamma^{\prime},g)$ follows from the non-
Archimedean property. ∎

######  Corollary 3.10.

Let $T$ and $T^{\prime}$ be normally embedded Hölder triangles with the
distance functions $f(x)=dist(x,T^{\prime})$ and
$g(x^{\prime})=dist(x^{\prime},T)$. For any $q\in\mathbb{F}$, the finite set
$L_{q}$ of maximal $q$-order zones $Z\subset V(T)$ for $f$ such that
$\mu(Z)<q$ is nonempty if, and only if, the set $L^{\prime}_{q}$ of maximal
$q$-order zones $Z^{\prime}\subset V(T^{\prime})$ for $g$ such that
$\mu(Z^{\prime})<q$ is nonempty, and there is a canonical one-to-one
correspondence $Z^{\prime}=\tau_{q}(Z)$ between the sets $L_{q}$ and
$L^{\prime}_{q}$ such that $tord(Z,\tau_{q}(Z))=q$.

###### Proof.

The finiteness of the set $L_{q}$ follows from the fact that, for a given
$q\in\mathbb{F}$, each pizza slice of a pizza on $T$ associated with $f$
contains at most one zone from $L_{q}$. ∎

######  Lemma 3.11.

Let $T$ and $T^{\prime}$ be normally embedded oriented Hölder triangles. Let
$Z\subset V(T)$ and $Z^{\prime}\subset V(T^{\prime})$ be maximal $q$-order
zones for $f$ and $g$ respectively, of orders $\mu(Z)=\mu(Z^{\prime})<q$,
related as in Proposition 3.9 and Corollary 3.10. Then the pairs
$(\tilde{T},\tilde{T}^{\prime})$ of Hölder triangles $\tilde{T}\subset T$ and
$\tilde{T}^{\prime}\subset T^{\prime}$ related as in Corollary 3.6, such that
$V(\tilde{T})\subset Z$ and $V(\tilde{T}^{\prime})\subset Z^{\prime}$, are
either all positively oriented or all negatively oriented.

###### Proof.

This follows from Remark 3.8, since for any two pairs of Hölder triangles in
Lemma 3.11 there is a larger pair of Hölder triangles containing both of them
and satisfying conditions of Corollary 3.6. ∎

######  Definition 3.12.

The pair of zones $Z\subset V(T)$ and $Z^{\prime}\subset V(T^{\prime})$ in
Lemma 3.11 is called _positively oriented_ (resp., _negatively oriented_) if
the pairs $(\tilde{T},\tilde{T}^{\prime})$ of Hölder triangles
$\tilde{T}\subset T$ and $\tilde{T}^{\prime}\subset T^{\prime}$ in Lemma 3.11
are positively oriented (resp., negatively oriented).

Figure 1. Illustration to the proof of Lemma 3.5. Figure 2. Illustration to
the proof of Lemma 3.13.

######  Lemma 3.13.

Let $T=T(\gamma_{1},\gamma_{2})$ and
$T^{\prime}=T(\gamma^{\prime}_{1},\gamma^{\prime}_{2})$ be two normally
embedded Hölder triangles, such that $T$ is elementary with respect to
$f(x)=dist(x,T^{\prime})$ and
$tord(\gamma_{1},\gamma^{\prime}_{1})=tord(T,T^{\prime})$. Then $T^{\prime}$
is elementary with respect to $g(x^{\prime})=dist(x^{\prime},T)$.

###### Proof.

We have to show that, for any Hölder triangle
$T^{\prime\prime}=T(\theta_{1},\theta_{2})\subset T^{\prime}$ such that
$tord(\theta_{1},T)=tord(\theta_{2},T)=q$, we have $tord(\gamma^{\prime},T)=q$
for each arc $\gamma^{\prime}\subset T^{\prime\prime}$. Let us show first that
$q^{\prime}=tord(\gamma^{\prime},T)\geq q$ for each arc
$\gamma^{\prime}\subset T^{\prime\prime}$. If $q^{\prime}<q$, let
$\lambda_{1}$ and $\lambda_{2}$ be arcs in $T$ such that
$tord(\lambda_{1},\theta_{1})=tord(\lambda_{2},\theta_{2})=q$. Lemma 3.5
implies that there is an arc $\lambda\subset T(\lambda_{1},\lambda_{2})$ such
that $tord(\lambda,T^{\prime})\leq q^{\prime}<q$, a contradiction with $T$
being elementary with respect to $f$.

Suppose now that $q^{\prime}>q$. We may assume that $\theta_{1}\subset
T(\gamma^{\prime}_{1},\gamma^{\prime})\subset T^{\prime}$. Since
$tord(\gamma_{1},\gamma^{\prime}_{1})=tord(T,T^{\prime})$, we have
$tord(\gamma_{1},\gamma^{\prime}_{1})\geq q^{\prime}$. Let $\gamma\subset T$
be an arc such that $tord(\gamma,\gamma^{\prime})=q^{\prime}$ (see Fig. 2).
Then Lemma 3.5 applied to $T(\gamma_{1},\gamma)\subset T$ and
$T(\gamma^{\prime}_{1},\gamma^{\prime})\subset T^{\prime}$ implies that there
is an arc $\lambda\subset T(\gamma_{1},\gamma)$ such that
$tord(\lambda,T^{\prime})\leq q$, a contradiction with $T$ being elementary
with respect to $f$. ∎

######  Corollary 3.14.

Let $T=T(\gamma_{1},\gamma_{2})$ and
$T^{\prime}=T(\gamma^{\prime}_{1},\gamma^{\prime}_{2})$ be normally embedded
Hölder triangles, such that $T$ is elementary with respect to
$f(x)=dist(x,T^{\prime})$ and
$tord(\gamma_{1},\gamma^{\prime}_{1})=tord(T,T^{\prime})$. Then, for any two
Hölder triangles $\tilde{T}=T(\gamma_{1},\lambda)\subset T$ and
$\tilde{T}^{\prime}=T(\gamma^{\prime}_{1},\lambda^{\prime})\subset
T^{\prime}$, $\tilde{T}$ is elementary with respect to
$\tilde{f}(x)=dist(x,\tilde{T}^{\prime})$ and $\tilde{T}^{\prime}$ is
elementary with respect to $\tilde{g}(x^{\prime})=dist(x^{\prime},\tilde{T})$.

###### Proof.

Since $\tilde{T}$ is elementary with respect to $f|_{\tilde{T}}$, Lemma 3.13
applied to $\tilde{T}$ instead of $T$ implies that $T^{\prime}$ is elementary
with respect to $h(x^{\prime})=dist(x^{\prime},\tilde{T})$. Thus
$\tilde{T}^{\prime}$ is elementary with respect to
$\tilde{g}=h|_{\tilde{T}^{\prime}}$. Lemma 3.13 applied to
$\tilde{T}^{\prime}$ instead of $T$ and $\tilde{T}$ instead of $T^{\prime}$
implies that $\tilde{T}$ is elementary with respect to $\tilde{f}$. ∎

######  Lemma 3.15.

Let $T=T(\gamma_{1},\gamma_{2})$ and
$T^{\prime}=T(\gamma^{\prime}_{1},\gamma^{\prime}_{2})$ be normally embedded
$\beta$-Hölder triangles satisfying (5). Suppose that $T$ is a pizza slice
associated with $f(x)=dist(x,T^{\prime})$. Then conditions of Proposition 3.2
are satisfied for $T$ and $T^{\prime}$. Moreover,
$\mu_{T,f}\equiv\mu_{T^{\prime},g}$, where $\mu_{T,f}(q)$ and
$\mu_{T^{\prime},g}(q)$ are the width functions defined on
$Q_{f}(T)=Q_{g}(T^{\prime})$.

###### Proof.

Since the five conditions of Proposition 3.2 are equivalent, it is enough to
prove condition 3: there is a bi-Lipschitz homeomorphism $h:T\to T^{\prime}$
such that $h(\gamma_{1})=\gamma^{\prime}_{1}$,
$h(\gamma_{2})=\gamma^{\prime}_{2}$ and
$tord(\gamma,h(\gamma))=tord(\gamma,T^{\prime})$ for each arc $\gamma\subset
T$.

Let $Q=Q_{f}(T)$, and let the width function
$\mu(q)=\mu_{T,f}(q):Q\to\mathbb{F}\cup\\{\infty\\}$ be affine, $\mu(q)=aq+b$.
We consider the following cases: (1) $Q=\\{\alpha\\}$ where $\alpha\leq\beta$,
(2) $Q=\\{\alpha\\}$ where $\alpha>\beta$, (3) $\mu(q)\equiv q$, (4)
$\mu(q)<q$ for all $q\in Q$, (5) $\mu(q)=q$ only for the maximal value of
$\mu(q)$, (6) $\mu(q)=q$ only for the minimal value $\mu(q)=\beta$.

Case 1. Any bi-Lipschitz homeomorphism $h:T\to T^{\prime}$ such that
$h(\gamma_{1})=\gamma^{\prime}_{1}$ and $h(\gamma_{2})=\gamma^{\prime}_{2}$
satisfies $tord(\gamma,h(\gamma))=tord(\gamma,T^{\prime})=\alpha$ for all arcs
$\gamma\in V(T)$.

Case 2. It follows from [7, Proposition 2.20] (see Proposition 3.3) that there
is a bi-Lipschitz homeomorphism $h:T\to T^{\prime}$ such that
$tord(\gamma,h(\gamma))\geq\alpha$ for any arc $\gamma\subset T$. Since
$tord(\gamma,h(\gamma))\leq tord(\gamma,T^{\prime})=\alpha$ for any arc
$\gamma\subset T$, we have $tord(\gamma,h(\gamma))=\alpha$.

Case 3. We may assume that $Q$ is not a point and
$q_{1}=tord(\gamma_{1},T^{\prime})$ is the maximal value of $q\in Q$. Then
$q=ord_{\gamma}f=\mu_{T}(\gamma,f)=tord(\gamma,\gamma_{1})$ for all arcs
$\gamma\subset T$ such that $tord(\gamma,\gamma_{1})\leq q_{1}$, otherwise
$ord_{\gamma}f=q_{1}$. Any bi-Lipschitz homeomorphism $h:T\to T^{\prime}$ such
that $h(\gamma_{1})=\gamma^{\prime}_{1}$ satisfies
$tord(h(\gamma),\gamma^{\prime}_{1})=tord(\gamma,\gamma_{1})$ for all arcs
$\gamma\subset T$. Thus
$q=ord_{\gamma}f=\mu_{T}(\gamma,f)=tord(\gamma,\gamma_{1})=tord(h(\gamma),\gamma^{\prime}_{1})$
for all $\gamma\subset T$ such that $tord(\gamma,\gamma_{1})\leq q_{1}$. Since
$tord(\gamma_{1},\gamma^{\prime}_{1})=q_{1}\geq q$, this implies that
$tord(\gamma,h(\gamma))\geq q$. If $tord(\gamma,h(\gamma))>q$ then
$tord(\gamma,T^{\prime})>q$, a contradiction. Thus $tord(\gamma,h(\gamma))=q$
for all $\gamma\subset T$ such that $tord(\gamma,\gamma_{1})\leq q_{1}$.
Otherwise, if $tord(\gamma,\gamma_{1})>q_{1}$, then
$tord(h(\gamma),\gamma^{\prime}_{1})>q_{1}$, thus
$tord(\gamma,h(\gamma))=tord(\gamma_{1},\gamma^{\prime}_{1})=q_{1}=tord(\gamma,T^{\prime})$.

Case 4. Using the same arguments as in the proof of [7, Proposition 2.20], we
assume that $T^{\prime}=T_{\beta}\subset\mathbb{R}^{2}$ is a standard
$\beta$-Hölder triangle (1), $T\cup T^{\prime}\subset\mathbb{R}^{n}$, and
$\pi:T\to\mathbb{R}^{2}$ is an orthogonal projection. We may also assume that
$Q$ is not a point, and that $\mu(q_{1})$, where
$q_{1}=tord(\gamma_{1},T^{\prime})$, is the maximal value of $\mu(q)$ for
$q\in Q$. Then $\mu_{T}(\gamma,f)=tord(\gamma,\gamma_{1})$ for all arcs
$\gamma\subset T$ such that $tord(\gamma,\gamma_{1})\leq\mu(q_{1})$, otherwise
$ord_{\gamma}f=q_{1}$.

The set $S\subset T$ where $\pi$ is not smooth and orientation-preserving is a
finite union of isolated arcs and $\beta_{j}$-Hölder triangles
$T_{j}=T(\lambda_{j},\lambda^{\prime}_{j})\subset T$. We want to show that $f$
has the same order $q_{j}$ on each arc $\gamma\subset T_{j}$. It is enough to
show that $ord_{\lambda_{j}}f=ord_{\lambda^{\prime}_{j}}f$. We may assume that
$\lambda^{\prime}_{j}\subset T(\gamma_{1},\lambda_{j})\subset T$, thus
$\mu_{j}=\mu_{T}(\lambda_{j},f)\leq\mu(\lambda^{\prime}_{j},f)$. If
$ord_{\lambda_{j}}f=q_{j}\neq ord_{\lambda^{\prime}_{j}}f$ then
$\beta_{j}\leq\mu_{j}=tord(\lambda_{j},\gamma_{1})$. Since $T_{j}$ is
orientation-reversing and $T$ is normally embedded, we have $\beta_{j}\geq
q_{j}$, a contradiction with the condition $\mu_{j}<q_{j}$. Thus
$ord_{\lambda_{j}}f=ord_{\lambda^{\prime}_{j}}f=q_{j}\leq\beta_{j}<\mu_{j}$,
and there is a $\mu_{j}$-Hölder triangle $\tilde{T}_{j}\subset T$ containing
$T_{j}$ such that $ord_{\gamma}f=q_{j}$ for each arc
$\gamma\subset\tilde{T}_{j}$.

It follows from [7, Proposition 2.20] that there is a bi-Lipschitz
orientation-preserving homeomorphism
$h_{j}:\tilde{T}_{j}\to\pi(\tilde{T}_{j})\cap T^{\prime}$ such that
$tord(\gamma,h_{j}(\gamma))=q_{j}$ for each arc $\gamma\subset\tilde{T}_{j}$.
One can choose triangles $\tilde{T}_{j}$ so that they are all disjoint.
Replacing projection $\pi$ with the homeomorphisms $h_{j}$ on each triangle
$\tilde{T}_{j}$, a bi-Lipschitz homeomorphism $h:T\to T^{\prime}$ can be
obtained, such that $tord(\gamma,h(\gamma))=tord(\gamma,T^{\prime})$ for each
arc $\gamma\subset T$.

Case 5. Assuming that $\mu(q_{1})=q_{1}=tord(\gamma_{1},\gamma^{\prime}_{1})$
is the maximal value of $\mu(q)$, for any arc $\gamma\subset T$ such that
$tord(\gamma,T^{\prime})=q_{1}$ we have
$tord(\gamma,\gamma_{1})\geq\mu(\gamma)=q_{1}$, thus
$tord(\gamma,h(\gamma))=tord(\gamma_{1},\gamma^{\prime}_{1})=q_{1}$. For any
arc $\gamma\subset T$ such that
$q=tord(\gamma,T^{\prime})>\mu(q)=tord(\gamma,\gamma_{1})$, the same arguments
as in Case 4 apply.

Case 6. The same arguments as in Case 4 imply that, for any triangle
$T_{j}\subset T$ containing an arc $\gamma$ such that $ord_{\gamma}f>\beta$
and $\pi|_{T}$ is orientation-reversing, the order $q_{j}$ of $f$ is the same
on all arcs of $T_{j}$, and there is a $\mu_{j}$-triangle $\tilde{T}_{j}$
containing $T_{j}$, where $\mu_{j}>\beta$, such that $\pi|_{\tilde{T}_{j}}$
can be replaced with a bi-Lipschitz orientation-preserving homeomorphism
$h_{j}:\tilde{T}_{j}\to\pi(\tilde{T}_{j})\cap T^{\prime}$, such that
$tord(\gamma,h_{j}(\gamma))=q_{j}$ for each arc $\gamma\subset\tilde{T}_{j}$.
This allows one to find $\beta$-Hölder triangles
$\tilde{T}=T(\gamma_{1},\tilde{\gamma})\subset T$ and
$\tilde{T}^{\prime}=T(\gamma^{\prime}_{1},\tilde{\gamma}^{\prime})\subset
T^{\prime}$ such that $\bar{T}=T(\tilde{\gamma},\gamma_{2})\subset T$ and
$\bar{T}^{\prime}=T(\tilde{\gamma}^{\prime},\gamma^{\prime}_{2})\subset
T^{\prime}$ are also $\beta$-Hölder triangles, $ord_{\gamma}f=\beta$ for each
arc $\gamma\subset\bar{T}$, and to obtain a bi-Lipschitz homeomorphism
$\tilde{h}:\tilde{T}\to\tilde{T}^{\prime}$ such that
$tord(\gamma,h(\gamma))=tord(\gamma,T^{\prime})$ for each arc
$\gamma\subset\tilde{T}$. After that, $\tilde{h}$ combined with any bi-
Lipschitz homeomorphism $\bar{h}:\bar{T}\to\bar{T}^{\prime}$, such that
$\bar{h}(\tilde{\gamma})=\tilde{\gamma}^{\prime}$ and
$\bar{h}(\gamma_{2})=\gamma^{\prime}_{2}$, defines a bi-Lipschitz
homeomorphism $h:T\to T^{\prime}$ such that
$tord(\gamma,h(\gamma))=tord(\gamma,T^{\prime})$ for each arc $\gamma\subset
T$.

The existence of a mapping $h:T\to T^{\prime}$ satisfying condition 5 of
Proposition 3.2 implies that $Q_{f}(T)=Q_{g}(T^{\prime})$ and
$\mu_{T,f}\equiv\mu_{T^{\prime},g}$. ∎

######  Definition 3.16.

Let $T=T(\gamma_{1},\gamma_{2})$ and
$T^{\prime}=T(\gamma^{\prime}_{1},\gamma^{\prime}_{2})$ be two normally
embedded oriented $\beta$-Hölder triangles satisfying (5), such that $T$ is a
pizza slice associated with $f(x)=dist(x,T^{\prime})$ and
$tord(T,T^{\prime})=tord(\gamma_{1},\gamma^{\prime}_{1})>\beta$. The pair
$(T,T^{\prime})$ is called _positively oriented_ if either $T$ is oriented
from $\gamma_{1}$ to $\gamma_{2}$ and $T^{\prime}$ from $\gamma^{\prime}_{1}$
to $\gamma^{\prime}_{2}$, or $T$ is oriented from $\gamma_{2}$ to $\gamma_{1}$
and $T^{\prime}$ from $\gamma^{\prime}_{2}$ to $\gamma^{\prime}_{1}$.
Otherwise, the pair $(T,T^{\prime})$ is called _negatively oriented_.

######  Lemma 3.17.

Let $T=T(\gamma_{1},\gamma_{2})$ and
$T^{\prime}=T(\gamma^{\prime}_{1},\gamma^{\prime}_{2})$ be two normally
embedded $\beta$-Hölder triangles in Definition 3.16 such that
$\mu_{T,f}(q)\not\equiv q$. For $q\in Q_{T}(f)$ such that $\mu(q)<q$, let
$Z_{q}\subset V(T)$ and $Z^{\prime}_{q}\subset V(T^{\prime})$ be the maximal
$q$-order zones for $f(x)=dist(x,T^{\prime})$ and
$g(x^{\prime})=dist(x^{\prime},T)$ respectively. Then the pair of zones
$(Z_{q},Z^{\prime}_{q})$ is positively oriented if the pair of Hölder
triangles $(T,T^{\prime})$ is positively oriented, and negatively oriented
otherwise.

######  Lemma 3.18.

Let a $\beta$-Hölder triangle $T=T(\gamma_{1},\gamma_{2})$ and a
$\beta^{\prime}$-Hölder triangle
$T^{\prime}=T(\gamma^{\prime}_{1},\gamma^{\prime}_{2})$ be normally embedded,
where
$\beta\geq\beta^{\prime},\;q_{1}=tord(\gamma_{1},\gamma^{\prime}_{1})=tord(T,T^{\prime})>\beta$
and $q_{2}=tord(\gamma_{2},T^{\prime})\geq\beta$. If $T$ is a pizza slice
associated with $f(x)=dist(x,T^{\prime})$ then there is an arc $\theta\subset
T^{\prime}$ such that $tord(\gamma^{\prime}_{1},\theta)=\beta$, and

(6) $tord(\gamma_{2},\theta)=tord(\theta,T)=q_{2},$

thus conditions (5) are satisfied for triangles $T$ and
$T(\gamma^{\prime}_{1},\theta)\subset T^{\prime}$. Moreover, if $q_{2}>\beta$
then $tord(\gamma^{\prime}_{1},\theta)=\beta$ for any arc $\theta\subset
T^{\prime}$ satisfying condition (6), and if $q_{2}=\beta$ then any arc
$\theta\subset T^{\prime}$ such that $tord(\gamma^{\prime}_{1},\theta)=\beta$
satisfies condition (6).

###### Proof.

Let $\theta\subset T^{\prime}$ be an arc such that
$tord(\gamma_{2},\theta)=q_{2}$. Note first that $\alpha=tord(\theta,T)\geq
q_{2}$. Suppose that $\alpha>q_{2}$, and let $\lambda\subset T$ be an arc such
that $tord(\lambda,\theta)=\alpha$,thus
$q^{\prime}=tord(\lambda,T^{\prime})\geq\alpha>q_{2}$. Let $\mu_{T}(q)$ be the
affine width function of $T$. Note that $\mu$ cannot be constant, since
$q_{1}\geq\alpha>q_{2}$, If the maximum of $\mu_{T}(q)$ is at $q=q_{2}$ then
$tord(\lambda,\gamma_{2})=\mu(q^{\prime})\leq\mu(\alpha)<\mu(q_{2})\leq
q_{2}$. Since $q_{2}=tord(\gamma_{2},\theta)$ and
$\alpha=tord(\lambda,\theta)>q_{2}$, this contradicts the non-Archimedean
property. Thus the maximum of $\mu_{T}(q)$ is at $q=q_{1}$ and its minimum is
$\mu(q_{2})=\beta$.

If $q_{2}>\beta$ then
$tord(\theta,\gamma^{\prime}_{1})=tord(\gamma_{1},\gamma_{2})=\beta$ for any
arc $\theta\subset T^{\prime}$ satisfying condition (6). However, since
$tord(\gamma_{1},\lambda)>\beta$,
$tord(\gamma_{1},\gamma^{\prime}_{1})=\mu(q_{1})>\beta$ and
$\alpha=tord(\lambda,\theta)>q_{2}\geq\beta$, condition
$tord(\gamma^{\prime}_{1},\theta)=\beta$ cannot be satisfied, a contradiction.
Thus $\alpha=q_{2}$ in this case.

Otherwise, if $q_{2}=\beta$, then any arc $\theta\subset T^{\prime}$ such that
$\alpha=tord(\theta,T)>\beta$ satisfies
$tord(\theta,\gamma^{\prime}_{1})\geq\min(\alpha,q_{1})>\beta$, thus any arc
$\theta\subset T^{\prime}$ such that $tord(\gamma^{\prime}_{1},\theta)=\beta$
satisfies condition (6). ∎

######  Proposition 3.19.

Let $T=T(\gamma_{1},\gamma_{2})$ and
$T^{\prime}=T(\gamma^{\prime}_{1},\gamma^{\prime}_{2})$ be normally embedded
$\beta$-Hölder triangles with the distance functions $f(x)=dist(x,T^{\prime})$
and $g(x^{\prime})=dist(x^{\prime},T)$, such that $T$ is elementary with
respect to $f$ and $tord(\gamma_{1},\gamma^{\prime}_{1})=tord(T,T^{\prime})$.
Let $\lambda\subset T$ and $\lambda^{\prime}\subset T^{\prime}$ be a regular
pair of arcs such that $\tilde{T}=T(\gamma_{1},\lambda)$ and
$\tilde{T}^{\prime}=T(\gamma^{\prime}_{1},\lambda^{\prime})$ are
$\tilde{\beta}$-Hölder triangles and $\check{T}=T(\lambda,\gamma_{2})$ and
$\check{T}^{\prime}=T(\lambda^{\prime},\gamma^{\prime}_{2})$ are
$\check{\beta}$-Hölder triangles. If both pairs
$(\tilde{T},\tilde{T}^{\prime})$ and $(\check{T},\check{T}^{\prime})$ satisfy
conditions of Proposition 3.2, then the pair $(T,T^{\prime})$ satisfies
conditions of Proposition 3.2.

###### Proof.

Since conditions 1 - 5 of Proposition 3.2 are equivalent, it is enough to
prove condition 3, i.e., to find a bi-Lipschitz homeomorphism $h:T\to
T^{\prime}$ such that $h(\gamma_{1})=\gamma^{\prime}_{1}$,
$h(\gamma_{2})=\gamma^{\prime}_{2}$ and
$tord(\gamma,h(\gamma))=tord(\gamma,T^{\prime})$ for each arc $\gamma\subset
T$. Conditions of Proposition 3.19 imply that there is a bi-Lipschitz
homeomorphism $\tilde{h}:\tilde{T}\to\tilde{T}^{\prime}$ such that
$\tilde{h}(\gamma_{1})=\gamma^{\prime}_{1}$,
$\tilde{h}(\lambda)=\lambda^{\prime}$ and
$tord(\gamma,\tilde{h}(\gamma))=tord(\gamma,\tilde{T}^{\prime})=tord(\tilde{h}(\gamma,\tilde{T})$
for each arc $\gamma\subset\tilde{T}$, and also a bi-Lipschitz homeomorphism
$\check{h}:\check{T}\to\check{T}^{\prime}$ such that
$\check{h}(\lambda)=\lambda^{\prime}$,
$\check{h}(\gamma_{2})=\gamma^{\prime}_{2}$ and
$tord(\gamma,\check{h}(\gamma))=tord(\gamma,\check{T}^{\prime})=tord(\check{h}(\gamma),\check{T})$
for each arc $\gamma\subset\check{T}$. We may assume that
$\tilde{h}(x)=\check{h}(x)$ for $x\in\lambda$.

We claim that a bi-Lipschitz homeomorphism $h:T\to T^{\prime}$ with the
necessary properties can be defined as $h(x)=\tilde{h}(x)$ for $x\in\tilde{T}$
and $h(x)=\check{h}(x)$ for $x\in\check{T}$. It is enough to show that
$tord(\gamma,T^{\prime})=tord(\gamma,\tilde{T}^{\prime})$ for any arc
$\gamma\subset\tilde{T}$ and
$tord(\gamma,T^{\prime})=tord(\gamma,\check{T}^{\prime})$ for any arc
$\gamma\subset\check{T}$.

Since $T$ is elementary with respect to $f$, Lemma 3.13 implies that
$T^{\prime}$ is elementary with respect to $g$. Corollary 3.14 implies that
$\tilde{T}$ is elementary with respect to $\tilde{f}$, $\tilde{T}^{\prime}$ is
elementary with respect to $\tilde{g}$, $\check{T}$ is elementary with respect
to $\check{f}$ and $\check{T}^{\prime}$ is elementary with respect to
$\check{g}$.

Let $\gamma\subset\tilde{T}$. Then
$\alpha=tord(\gamma,T^{\prime})=max(tord(\gamma,\tilde{T}^{\prime}),tord(\gamma,\check{T}^{\prime})\geq
tord(\gamma,\tilde{T}^{\prime})$. If $\alpha>tord(\gamma,\tilde{T}^{\prime})$
then there is an arc $\gamma^{\prime}\subset\check{T}^{\prime}$ such that
$tord(\gamma,\gamma^{\prime})=tord(\gamma,\check{T}^{\prime})=\alpha$, thus
$tord(\gamma^{\prime},T)\geq\alpha$. Then

$\alpha>tord(\gamma,\tilde{h}(\gamma))=tord(\tilde{h}(\gamma),\tilde{T})\geq
tord(\lambda^{\prime},\tilde{T})=tord(\lambda,\lambda^{\prime})=tord(\lambda^{\prime},\check{T})$

Thus $tord(\gamma^{\prime},T)>tord(\lambda^{\prime},T)$, a contradiction with
$T^{\prime}$ being elementary with respect to $g$.

Let $\gamma\subset\check{T}$. Then $\alpha=tord(\gamma,T^{\prime})\leq
tord(\lambda,T^{\prime})=tord(\lambda,\lambda^{\prime})$. If
$\alpha>tord(\gamma,\check{T}^{\prime})$ then there is an arc
$\gamma^{\prime}\subset\tilde{T}^{\prime}$ such that
$tord(\gamma,\gamma^{\prime})=\alpha>tord(\gamma,\check{T}^{\prime}\geq
tord(\gamma,\lambda^{\prime})$ (see Fig. 3). This implies that
$tord(\gamma^{\prime},\lambda^{\prime})=tord(\gamma,\lambda^{\prime})$. Since
$tord(\tilde{h}^{-1}(\gamma^{\prime}),\gamma)\leq
tord(\tilde{h}^{-1}(\gamma^{\prime}),\lambda)=tord(\gamma^{\prime},\lambda^{\prime})$,
we have
$tord(\tilde{h}^{-1}(\gamma^{\prime}),\gamma^{\prime})=tord(\tilde{h}^{-1}(\gamma^{\prime}),\gamma)<\alpha$.
Since $T$ is elementary with respect to $f$, we have
$tord(\tilde{h}^{-1}(\gamma^{\prime}),\gamma^{\prime})\geq
tord(\lambda,\lambda^{\prime})\geq\alpha$, a contradiction with condition
$tord(\tilde{h}^{-1}(\gamma^{\prime}),\gamma)<\alpha$. Thus
$tord(\gamma,T^{\prime})=tord(\gamma,\check{T})$. ∎

Figure 3. Illustration to the proof of Proposition 3.19.

######  Theorem 3.20.

Let $T=T(\gamma_{1},\gamma_{2})$ and
$T^{\prime}=T(\gamma^{\prime}_{1},\gamma^{\prime}_{2})$ be normally embedded
$\beta$-Hölder triangles satisfying (5). Let $f(x)=dist(x,T^{\prime})$ for
$x\in T$ and $g(x^{\prime})=dist(x^{\prime},T)$ for $x^{\prime}\in
T^{\prime}$. If $T$ is elementary with respect to $f$ then $T$ and
$T^{\prime}$ satisfy conditions of Proposition 3.2. Moreover,
$\mu_{T,f}\equiv\mu_{T^{\prime},g}$, where $\mu_{T,f}(q)$ and
$\mu_{T^{\prime},g}(q)$ are the width functions defined on
$Q_{f}(T)=Q_{g}(T^{\prime})$.

###### Proof.

Since the triangles are elementary, we may assume that
$tord(\gamma_{1},\gamma^{\prime}_{1})=tord(T,T^{\prime})$. Let
$\\{T_{i}\\}_{i=1}^{p}$ be a minimal pizza decomposition of $T$ associated
with $f$, where each pizza slice $T_{i}=T(\lambda_{i-1},\lambda_{i})$ is a
$\beta_{i}$-Hölder triangle, $\lambda_{0}=\gamma_{1}$ and
$\lambda_{p}=\gamma_{2}$. We proceed by induction on the number $p$ of pizza
slices. The case $p=1$ follows from Lemma 3.15. If $p>1$ then
$tord(\lambda_{1},T^{\prime})>\beta$, otherwise $\\{T_{i}\\}$ would not be a
minimal pizza decomposition. It follows from Lemma 3.18 applied to $T_{1}$
that there is an arc $\theta_{1}\subset T^{\prime}$ such that
$tord(\gamma^{\prime}_{1},\theta)=\beta_{1}$ and conditions of Proposition 3.2
are satisfied for $T_{1}$ and
$T^{\prime}_{1}=T(\gamma^{\prime}_{1},\theta_{1})$.

Let $\check{T}=T(\lambda_{1},\gamma_{2})$ and
$\check{T}^{\prime}=T(\theta_{1},\gamma^{\prime}_{2})$. Since
$tord(\lambda_{1},T^{\prime})>\beta$, $\check{T}$ and $\check{T}^{\prime}$
have the same exponents (see Remark 3.4). The same arguments as in the proof
of Proposition 3.19 show that
$tord(\gamma,\check{T}^{\prime})=tord(\gamma,T^{\prime})$ for any arc
$\gamma\subset\check{T}$, thus $\\{T_{i}\\}_{i=2}^{p}$ is a minimal pizza
decomposition of $\check{T}$ associated with the function
$\check{f}(x)=dist(x,\check{T}^{\prime})$. By the inductional hypothesis,
Hölder triangles $\check{T}$ and $\check{T}^{\prime}$ satisfy conditions of
Proposition 3.2. Proposition 3.19 implies that $T=T_{1}\cup\check{T}$ and
$T^{\prime}=\mathcal{T}^{\prime}_{1}\cup\check{T}^{\prime}$ satisfy conditions
of Proposition 3.2. The existence of a mapping $h:T\to T^{\prime}$ satisfying
condition 5 of Proposition 3.2 implies that $Q_{f}(T)=Q_{g}(T^{\prime})$ and
$\mu_{T,f}\equiv\mu_{T^{\prime},g}$. ∎

##  4\. The $\sigma\tau$-pizza invariant.

If two normally embedded Hölder triangles $T=T(\gamma_{1},\gamma_{2})$ and
$T^{\prime}=T(\gamma^{\prime}_{1},\gamma^{\prime}_{2})$ satisfying condition
(5) are not elementary with respect to the distance functions, then $T\cup
T^{\prime}$ may be not outer bi-Lipschitz equivalent to the union of $T$ and a
graph of a function defined on $T$ (see Fig. 4). In any case, a minimal pizza
on $T$ associated with the function $f(x)=dist(x,T^{\prime})$, and a minimal
pizza on $T^{\prime}$ associated with the function
$g(x^{\prime})=dist(x^{\prime},T)$, are outer Lipschitz invariants of the pair
$(T,T^{\prime})$. The following example shows that two pairs $(T,T^{\prime})$
and $(\tilde{T},\tilde{T}^{\prime})$ of normally embedded triangles satisfying
condition (5) may be not outer bi-Lipschitz equivalent even when the minimal
pizzas on $T$ and $T^{\prime}$ are equivalent to the minimal pizzas on
$\tilde{T}$ and $\tilde{T}^{\prime}$ respectively.

######  Example 4.1.

The links of two normally embedded Hölder triangles
$T=T(\gamma_{1},\gamma_{2})$ and
$T^{\prime}=T(\gamma^{\prime}_{1},\gamma^{\prime}_{2})$ are shown in Fig. 5.
Triangle $T$ is partitioned by the arcs $\lambda_{1}$, $\lambda_{2}$,
$\lambda_{3}$, $\lambda_{4}$ into Hölder triangles
$T_{1}=T(\gamma_{1},\lambda_{1})$, $T_{2}=T(\lambda_{1},\lambda_{2})$,
$T_{3}=T(\lambda_{2},\lambda_{3})$, $T_{4}=T(\lambda_{3},\lambda_{4})$,
$T_{5}=T(\lambda_{4},\gamma_{2})$ with exponents $\mu_{2}$, $q_{2}$,
$\mu_{1}$, $q_{2}$, $\mu_{2}$, respectively, and triangle $T^{\prime}$ is
partitioned by the arcs $\lambda^{\prime}_{1}$, $\lambda^{\prime}_{2}$,
$\lambda^{\prime}_{3}$, $\lambda^{\prime}_{4}$ into Hölder triangles
$T^{\prime}_{1}=T(\gamma^{\prime}_{1},\lambda^{\prime}_{1})$,
$T^{\prime}_{2}=T(\lambda^{\prime}_{1},\lambda^{\prime}_{2})$,
$T^{\prime}_{3}=T(\lambda^{\prime}_{2},\lambda^{\prime}_{3})$,
$T^{\prime}_{4}=T(\lambda^{\prime}_{3},\lambda^{\prime}_{4})$,
$T^{\prime}_{5}=T(\lambda^{\prime}_{4},\gamma^{\prime}_{2})$ with exponents
$\mu_{2}$, $q_{2}$, $\mu_{1}$, $q_{2}$, $\mu_{2}$, respectively, so that the
following holds:
$tord(\gamma,T^{\prime})=q_{2}$ for any arc $\gamma\subset T_{1}$,
$tord(\gamma,T^{\prime})=q_{1}$ for any arc $\gamma\subset T_{3}$,
$tord(\gamma,T^{\prime})=q_{2}$ for any arc $\gamma\subset T_{5}$;
$tord(\gamma,T^{\prime})=tord(\gamma,\lambda^{\prime}_{2})$ for any arc
$\gamma\subset T_{2}$,
$tord(\gamma,T^{\prime})=tord(\gamma,\lambda^{\prime}_{3})$ for any arc
$\gamma\subset T_{4}$; $tord(\gamma^{\prime},T)=q_{2}$ for any arc
$\gamma^{\prime}\subset T^{\prime}_{1}$, $tord(\gamma^{\prime},T)=q_{1}$ for
any arc $\gamma^{\prime}\subset T^{\prime}_{3}$,
$tord(\gamma^{\prime},T)=q_{2}$ for any arc $\gamma^{\prime}\subset
T^{\prime}_{5}$; $tord(\gamma^{\prime},T)=tord(\gamma^{\prime},\lambda_{2})$
for any arc $\gamma^{\prime}\subset T^{\prime}_{2}$,
$tord(\gamma^{\prime},T)=tord(\gamma^{\prime},\lambda_{3})$ for any arc
$\gamma^{\prime}\subset T^{\prime}_{4}$.

In particular, $T$ and $T^{\prime}$ satisfy condition (5):

$tord(\gamma_{1},T^{\prime})=tord(\gamma_{1},\gamma^{\prime}_{1})=tord(\gamma^{\prime}_{1},T)=tord(\gamma_{2},T^{\prime})=tord(\gamma_{2},\gamma^{\prime}_{2})=tord(\gamma^{\prime}_{2},T)=q_{2}.$

Assuming $q_{1}>\mu_{1}\geq q_{2}>\mu_{2}$, the arcs
$\lambda_{1},\ldots,\lambda_{4}$ define a minimal pizza decomposition of $T$
associated with the function $f(x)=dist(x,T^{\prime})$.

Although $T$ is not elementary with respect to $f(x)$, the union $T\cup
T^{\prime}$ is outer bi-Lipschitz equivalent to the union of $T$ and the graph
of $f(x)$. In particular, a minimal pizza on $T^{\prime}$ associated with the
function $g(x^{\prime})=dist(x^{\prime},T)$ is equivalent to a minimal pizza
on $T$ associated with the function $f(x)$.

Hölder triangles $\tilde{T}$ and $\tilde{T}^{\prime}$ in Fig. 6 are also
normally embedded and satisfy condition (5):

$tord(\gamma_{1},\tilde{T}^{\prime})=tord(\gamma_{1},\gamma^{\prime}_{1})=tord(\gamma^{\prime}_{1},\tilde{T})=tord(\gamma_{2},\tilde{T}^{\prime})=tord(\gamma_{2},\gamma^{\prime}_{2})=tord(\gamma^{\prime}_{2},\tilde{T})=q_{2}.$

Triangle $\tilde{T}$ is partitioned by the arcs $\lambda_{1}$, $\lambda_{2}$,
$\lambda_{3}$, $\lambda_{4}$ into Hölder triangles
$\tilde{T}_{1}=T(\gamma_{1},\lambda_{1})$,
$\tilde{T}_{2}=T(\lambda_{1},\lambda_{2})$,
$\tilde{T}_{3}=T(\lambda_{2},\lambda_{3})$,
$\tilde{T}_{4}=T(\lambda_{3},\lambda_{4})$,
$\tilde{T}_{5}=T(\lambda_{4},\gamma_{2})$, and triangle $\tilde{T}^{\prime}$
is partitioned by the arcs $\lambda^{\prime}_{1}$, $\lambda^{\prime}_{2}$,
$\lambda^{\prime}_{3}$, $\lambda^{\prime}_{4}$ into Hölder triangles
$\tilde{T}^{\prime}_{1}=T(\gamma^{\prime}_{1},\lambda^{\prime}_{1})$,
$\tilde{T}^{\prime}_{2}=T(\lambda^{\prime}_{1},\lambda^{\prime}_{2})$,
$\tilde{T}^{\prime}_{3}=T(\lambda^{\prime}_{2},\lambda^{\prime}_{3})$,
$\tilde{T}^{\prime}_{4}=T(\lambda^{\prime}_{3},\lambda^{\prime}_{4})$,
$\tilde{T}^{\prime}_{5}=T(\lambda^{\prime}_{4},\gamma^{\prime}_{2})$.
Conditions satisfied by these triangles are the same as for those in Fig. 5,
except $tord(\gamma,\tilde{T}^{\prime})=tord(\gamma,\lambda^{\prime}_{3})$ for
any arc $\gamma\subset\tilde{T}_{2}$,
$tord(\gamma,\tilde{T}^{\prime})=tord(\gamma,\lambda^{\prime}_{2})$ for any
arc $\gamma\subset\tilde{T}_{4}$;
$tord(\gamma^{\prime},\tilde{T})=tord(\gamma^{\prime},\lambda_{3})$ for any
arc $\gamma^{\prime}\subset\tilde{T}^{\prime}_{2}$,
$tord(\gamma^{\prime},\tilde{T})=tord(\gamma^{\prime},\lambda_{2})$ for any
arc $\gamma^{\prime}\subset\tilde{T}^{\prime}_{4}$. Assuming
$q_{1}>\mu_{1}\geq q_{2}>\mu_{2}$, the arcs $\lambda_{1},\ldots,\lambda_{4}$
define a minimal pizza decomposition of $\tilde{T}$ associated with the
function $\tilde{f}(x)=dist(x,\tilde{T}^{\prime})$.

One can show that minimal pizzas on $T$ and $\tilde{T}$ are equivalent, and a
minimal pizza on $\tilde{T}^{\prime}$ associated with the function
$\tilde{g}(x^{\prime})=dist(x^{\prime},\tilde{T})$ is equivalent to a minimal
pizza on $\tilde{T}$ associated with the function $\tilde{f}(x)$. Thus minimal
pizzas on $T^{\prime}$ and $\tilde{T}^{\prime}$ are also equivalent.

However, the union $\tilde{T}\cup\tilde{T}^{\prime}$ is not outer bi-Lipschitz
equivalent to the union of $\tilde{T}$ and the graph of $\tilde{f}(x)$. In
particular, $\tilde{T}\cup\tilde{T}^{\prime}$ is not outer bi-Lipschitz
equivalent to $T\cup T^{\prime}$.

Figure 4. Two normally embedded $\beta$-Hölder triangles, not elementary with
respect to the distance functions. Shaded disks indicate zones with the
tangency order higher than $\beta$. Figure 5. Two normally embedded Hölder
triangles $T$ and $T^{\prime}$ in Example 4.1.

Figure 6. Two normally embedded Hölder triangles $\tilde{T}$ and
$\tilde{T}^{\prime}$ in Example 4.1.

######  Definition 4.2.

Let $T=T(\gamma_{1},\gamma_{2})$ and
$T^{\prime}=T(\gamma^{\prime}_{1},\gamma^{\prime}_{2})$ be normally embedded
$\beta$-Hölder triangles, oriented from $\gamma_{1}$ to $\gamma_{2}$ and from
$\gamma^{\prime}_{1}$ to $\gamma^{\prime}_{2}$ respectively, satisfying
condition (5). Let $f(x)=dist(x,T^{\prime})$ and
$g(x^{\prime})=dist(x^{\prime},T)$ be the distance functions defined on $T$
and $T^{\prime}$ respectively. Let $D_{\ell}\subset V(T)$, for
$\ell=0,\ldots,p$, be the pizza zones of a minimal pizza on $T$ associated
with $f(x)$, ordered according to the orientation of $T$, and let
$q_{\ell}=tord(D_{\ell},T^{\prime})=ord_{\gamma}f$ for any arc $\gamma\in
D_{\ell}$. A zone $D_{\ell}$ is called a _maximal exponent zone_ for $f(x)$
(or simply a _maximum zone_) if either $0<\ell<p$ and
$q_{\ell}\geq\max(q_{\ell-1},q_{\ell+1})$, or $\ell=0$ and $\beta<q_{0}\geq
q_{1}$, or $\ell=p$ and $\beta<q_{p}\geq q_{p-1}$. If a zone $D_{\ell}$ is not
a maximum zone, it is called a _minimal exponent zone_ for $f(x)$ (or simply a
_minimum zone_) if either $0<\ell<p$ and
$q_{\ell}\leq\min(q_{\ell-1},q_{\ell+1})$, or $\ell=0$ and $q_{0}\leq q_{1}$,
or $\ell=p$ and $q_{p}\leq q_{p-1}$. Maximum and minimum pizza zones
$D^{\prime}_{\ell^{\prime}}\subset V(T^{\prime})$ for a minimal pizza on
$T^{\prime}$ associated with $g(x^{\prime})$ are defined similarly, exchanging
$T$ and $T^{\prime}$.

######  Remark 4.3.

Each of the singular pizza zones $D_{0}=\\{\gamma_{1}\\}$ and
$D_{p}=\\{\gamma_{2}\\}$ is either a maximum or a minimum zone. When $p=1$ and
$q_{0}=q_{1}>\beta$, both $D_{0}$ and $D_{1}$ are maximum zones. When $p=1$
and $q_{0}=q_{1}\leq\beta$, both $D_{0}$ and $D_{1}$ are minimum zones. If
$p>1$ and $0<\ell<p$, then $q_{\ell}>\min(q_{\ell-1},q_{\ell+1})$ if
$D_{\ell}$ is a maximum zone, $q_{\ell}<\max(q_{\ell-1},q_{\ell+1})$ if
$D_{\ell}$ is a minimum zone.

######  Proposition 4.4.

Let $T=T(\gamma_{1},\gamma_{2})$ and
$T^{\prime}=T(\gamma^{\prime}_{1},\gamma^{\prime}_{2})$ be two normally
embedded Hölder triangles, oriented from $\gamma_{1}$ to $\gamma_{2}$ and from
$\gamma^{\prime}_{1}$ to $\gamma^{\prime}_{2}$ respectively, satisfying
condition (5). Let $\\{M_{i}\\}_{i=1}^{m}$ and
$\\{M^{\prime}_{i^{\prime}}\\}_{i^{\prime}=1}^{m^{\prime}}$ be the maximum
zones in $V(T)$ and $V(T^{\prime})$ for the distance functions
$f(x)=dist(x,T^{\prime})$ and $g(x^{\prime})=dist(x^{\prime},T)$ respectively,
ordered according to the orientations of $T$ and $T^{\prime}$. Let
$\bar{q}_{i}=tord(M_{i},T^{\prime})$ and
$\bar{q}^{\prime}_{i^{\prime}}=tord(M^{\prime}_{i^{\prime}},T)$. Then
$m^{\prime}=m$, and there is a canonical one-to-one correspondence
$i^{\prime}=\sigma(i)$ between the zones $M_{i}$ and
$M^{\prime}_{i^{\prime}}$, such that $\mu(M^{\prime}_{i^{\prime}})=\mu(M_{i})$
and
$tord(M_{i},M^{\prime}_{i^{\prime}})=\bar{q}_{i}=\bar{q}^{\prime}_{i^{\prime}}$.
If $\\{\gamma_{1}\\}=M_{1}$ is a maximum zone then
$M^{\prime}_{1}=\\{\gamma^{\prime}_{1}\\}$ and $\sigma(1)=1$. If
$\\{\gamma_{2}\\}=M_{m}$ is a maximum zone then
$M^{\prime}_{m}=\\{\gamma^{\prime}_{2}\\}$ and $\sigma(m)=m$.

###### Proof.

The case $p=1$ follows from Lemma 3.15, thus we may assume $p>1$.

Let us choose any arcs $\lambda_{i}\in D_{i}$, so that
$\\{T_{i}=T(\lambda_{i-1},\lambda_{i})\\}$ is a minimal pizza on $T$
associated with the function $f(x)$. Let $q_{i}=ord_{\lambda_{i}}f$ for
$i=0,\ldots,p$, and let $\beta_{i}=tord(\lambda_{i-1},\lambda_{i})$ be the
exponent of a pizza slice $T_{i}$, for $i=1,\ldots,p$.

Consider first the case when $\\{\gamma_{1}\\}=\\{\lambda_{0}\\}$ is a maximum
zone for $f(x)$. Then $q_{0}=tord(\gamma_{1},\gamma^{\prime}_{1})$ and
$q_{1}=ord_{\lambda_{1}}f\leq q_{0}$. If $q_{1}>\beta_{1}$, it follows from
Lemma 3.15 that, for any arc $\lambda^{\prime}\subset T^{\prime}$ such that
$tord(\lambda_{1},\lambda^{\prime})=q_{1}$, Hölder triangles $T_{1}$ and
$T^{\prime}_{1}=T(\gamma^{\prime}_{1},\lambda^{\prime})$ satisfy (5) and
conditions of Proposition 3.2. If $\gamma^{\prime}_{1}$ is not a maximum zone
for $g(x^{\prime})$ then, for any arc $\lambda^{\prime}_{1}$ such that
$T^{\prime}_{1}=T(\gamma^{\prime}_{1},\lambda^{\prime}_{1})$ is a pizza slice
for a minimal pizza associated with $g(x^{\prime})$, we have
$q^{\prime}_{1}=tord(\lambda^{\prime}_{1},T)>q_{0}$. Let $\lambda\subset T$ be
any arc such that $tord(\lambda,\lambda^{\prime}_{1})=q^{\prime}_{1}$. Then
Lemma 3.15 applied to $T^{\prime}_{1}$ and $\bar{T}=T(\gamma_{1},\lambda)$
implies that $\bar{T}$ is a pizza slice for $f(x)$. Since
$tord(\gamma,T^{\prime})\leq q_{0}$ for any arc $\gamma\subset T_{1}$, we have
$T_{1}\subset\bar{T}$, a contradiction with $T_{1}$ being a pizza slice for a
minimal pizza associated with $f(x)$.

Similarly, if $\\{\gamma_{2}\\}$ is a maximum zone for $f(x)$ then
$\\{\gamma^{\prime}_{2}\\}$ is a maximum zone for $g(x^{\prime})$.

Suppose next that $p>1$ and $M_{i}$ is a maximum zone for $f(x)$, where
$0<i<p$. Let $\lambda^{\prime}\subset T^{\prime}$ be any arc such that
$tord(\lambda_{i},\lambda^{\prime})=q_{i}$. We are going to show that
$\lambda^{\prime}$ belongs to a maximum zone $D^{\prime}$ for a minimal pizza
on $T^{\prime}$ associated with $g(x^{\prime})$.

Note first that, if $\lambda^{\prime}$ belongs to a pizza zone $D^{\prime}$
for a minimal pizza on $T^{\prime}$ associated with $g(x^{\prime})$, the same
arguments as those for $\gamma_{1}$ and $\gamma^{\prime}_{1}$ show that
$D^{\prime}$ is a maximum zone for $g(x^{\prime})$.

Suppose that $\lambda^{\prime}$ does not belong to a pizza zone. Let
$D^{\prime}_{j-1}$ and $D^{\prime}_{j}$ be two adjacent pizza zones for a
minimal pizza on $T^{\prime}$ associated with $g(x^{\prime})$,
$\lambda^{\prime}_{j-1}\in D^{\prime}_{j-1}$ and $\lambda^{\prime}_{j}\in
D^{\prime}_{j}$, $\lambda^{\prime}\subset
T^{\prime}_{j}=T(\lambda^{\prime}_{j-1},\lambda^{\prime}_{j})$ and
$\lambda^{\prime}\notin D^{\prime}_{j-1}\cup D^{\prime}_{j}$. The same
arguments as those for $\gamma_{1}$ and $\gamma^{\prime}_{1}$ show that
$tord(\lambda^{\prime}_{j-1},T)\leq q_{i}$ and
$tord(\lambda^{\prime}_{j},T)\leq q_{i}$. Since $T^{\prime}_{j}$ is a pizza
slice, at least one of these inequalities is an equality, say
$tord(\lambda^{\prime}_{j},T)=q_{i}$. Let
$\mu^{\prime}_{j}=\nu(\lambda^{\prime}_{j})\leq q_{i}$ be the order of the
pizza zone $D^{\prime}_{j}$. Since any arc $\gamma^{\prime}\subset
T^{\prime}_{j}$ such that
$tord(\gamma^{\prime},\lambda^{\prime}_{j})\geq\nu(\lambda^{\prime}_{j})$
belongs to $D^{\prime}_{j}$ and $\lambda^{\prime}\notin D^{\prime}_{j}$, we
have $tord(\lambda^{\prime},\lambda^{\prime}_{j})<\mu^{\prime}_{j}\leq q_{i}$.
Thus $\bar{T}^{\prime}=T(\lambda^{\prime},\lambda^{\prime}_{j})$ is a
$\bar{\beta}$-Hölder triangle, where $\bar{\beta}<q_{i}$, such that
$tord(\gamma^{\prime},T)=q_{i}$ for any arc
$\gamma^{\prime}\subset\bar{T}^{\prime}$. ∎

######  Definition 4.5.

_Let $T=T(\gamma_{1},\gamma_{2})$ and
$T^{\prime}=T(\gamma^{\prime}_{1},\gamma^{\prime}_{2})$ be two normally
embedded Hölder triangles, oriented from $\gamma_{1}$ to $\gamma_{2}$ and from
$\gamma^{\prime}_{1}$ to $\gamma^{\prime}_{2}$ respectively, satisfying
condition (5). Let $\\{M_{i}\\}_{i=1}^{m}$ and
$\\{M^{\prime}_{i^{\prime}}\\}_{i^{\prime}=1}^{m^{\prime}}$ be the maximum
zones in $V(T)$ and $V(T^{\prime})$ for the functions
$f(x)=dist(x,T^{\prime})$ and $g(x^{\prime})=dist(x^{\prime},T)$ respectively,
ordered according to the orientations of $T$ and $T^{\prime}$. According to
Proposition 4.4, we have $m^{\prime}=m$, and there is a canonical permutation
$\sigma$ of the set $\\{1,\ldots,m\\}$, the _characteristic permutation_ of
the pair $T$ and $T^{\prime}$, such that
$tord(M_{i},M^{\prime}_{\sigma(i)})=tord(M_{i},T^{\prime})=tord(M^{\prime}_{\sigma(i)},T)$._

######  Definition 4.6.

Let $T=T(\gamma_{1},\gamma_{2})$ and
$T^{\prime}=T(\gamma^{\prime}_{1},\gamma^{\prime}_{2})$ be two normally
embedded Hölder triangles, oriented from $\gamma_{1}$ to $\gamma_{2}$ and from
$\gamma^{\prime}_{1}$ to $\gamma^{\prime}_{2}$ respectively, satisfying
condition (5). Let $D_{\ell}\subset V(T)$, for $\ell=0,\ldots,p$, be the pizza
zones of a minimal pizza $\\{T_{\ell}=T(\lambda_{\ell-1},\lambda_{\ell})\\}$
on $T$ associated with the distance function $f(x)=dist(x,T^{\prime})$,
ordered according to the orientation of $T$. For $\ell=1,\ldots,p$, let
$Y_{\ell}=D_{\ell-1}\cup D_{\ell}\cup V(T_{\ell})$ be the maximal pizza slice
zones in $V(T)$ associated with $f$ (see Corollary 2.31). Let
$Q_{\ell}=Q_{f}(Y_{\ell})=[q_{\ell-1},q_{\ell}]$, where
$q_{\ell}=ord_{\lambda_{\ell}}f$, and
$\mu_{\ell}=\mu_{Y_{\ell},f}:Q_{\ell}\to\mathbb{F}\cup\\{\infty\\}$ be the
corresponding exponent intervals and affine width functions (see Definition
2.12). We say that a zone $Y_{\ell}$, and a pizza slice
$T_{\ell}=T(\lambda_{\ell-1},\lambda_{\ell})$ where $\lambda_{\ell-1}\in
D_{\ell-1}$ and $\lambda_{\ell}\in D_{\ell}$, is _transversal_ if
$\mu_{\ell}(q)\equiv q$, and _non-transversal_ otherwise.

######  Proposition 4.7.

Let $T=T(\gamma_{1},\gamma_{2})$ and
$T^{\prime}=T(\gamma^{\prime}_{1},\gamma^{\prime}_{2})$ be two normally
embedded Hölder triangles, oriented from $\gamma_{1}$ to $\gamma_{2}$ and from
$\gamma^{\prime}_{1}$ to $\gamma^{\prime}_{2}$ respectively, satisfying
condition (5). Let $D_{\ell}$, $Y_{\ell}$, $Q_{\ell}=[q_{\ell-1},q_{\ell}]$
and $\mu_{\ell}$ be as in Definition 4.6. Let $D^{\prime}_{\ell^{\prime}}$,
for $\ell^{\prime}=0,\ldots,p^{\prime}$, be the pizza zones of a minimal pizza
on $T^{\prime}$ associated with $g(x^{\prime})=dist(x^{\prime},T)$, ordered
according to the orientation of $T^{\prime}$. Let
$Y^{\prime}_{\ell^{\prime}}\subset V(T^{\prime})$,
$Q^{\prime}_{\ell^{\prime}}=Q_{g}(Y^{\prime}_{\ell^{\prime}})=[q_{\ell^{\prime}-1},q_{\ell^{\prime}}]\subset\mathbb{F}\cup\\{\infty\\}$
and
$\mu^{\prime}_{\ell^{\prime}}:Q^{\prime}_{\ell^{\prime}}\to\mathbb{F}\cup\\{\infty\\}$
be the corresponding maximal pizza slice zones, exponent intervals and affine
width functions. Then, for each index $\ell$ such that the pizza slice zone
$Y_{\ell}$ is non-transversal, there is a unique index
$\ell^{\prime}=\tau(\ell)$ such that $Q^{\prime}_{\ell^{\prime}}=Q_{\ell}$,
$\mu^{\prime}_{\ell^{\prime}}\equiv\mu_{\ell}$ and one of the following two
conditions holds:

(7)
$tord(D_{\ell},T^{\prime})=tord(D_{\ell},D^{\prime}_{\ell^{\prime}})=tord(D^{\prime}_{\ell^{\prime}},T),\\\
tord(D_{\ell-1},T^{\prime})=tord(D_{\ell-1},D^{\prime}_{\ell^{\prime}-1})=tord(D^{\prime}_{\ell^{\prime}-1},T);$
(8)
$tord(D_{\ell},T^{\prime})=tord(D_{\ell},D^{\prime}_{\ell^{\prime}-1})=tord(D^{\prime}_{\ell^{\prime}-1},T),\\\
tord(D_{\ell-1},T^{\prime})=tord(D_{\ell-1},D^{\prime}_{\ell^{\prime}})=tord(D^{\prime}_{\ell^{\prime}},T).$

###### Proof.

Let $Y_{\ell}\subset V(T)$ be a non-transversal maximal pizza slice zone for a
minimal pizza associated with $f$. For each $q\in Q_{\ell}$ let $Z_{q}\subset
Y_{\ell}$ be the maximal $q$-order zone for $f$. If $Q_{\ell}=\\{q_{\ell}\\}$
is a point, then $q_{\ell}>\mu_{\ell}$ since $Y_{\ell}$ is non-transversal. It
follows from Proposition 3.9 and Corollary 3.10 that there is a unique maximal
$q_{\ell}$-order zone $Z^{\prime}\subset V(T^{\prime})$ for $g$, of order
$\mu_{\ell}$, containing all arcs $\gamma^{\prime}\subset V(T^{\prime})$ such
that $tord(\gamma^{\prime},Y_{\ell})=ord_{\gamma^{\prime}}g=q_{\ell}$. Let us
show that $Z^{\prime}$ is a maximal pizza slice zone for $g$. Let
$Z^{\prime}\subset Y^{\prime}_{\ell^{\prime}}$ where
$Y^{\prime}_{\ell^{\prime}}$ is a maximal pizza slice zone for a minimal pizza
associated with $g$, of order $\mu^{\prime}_{\ell^{\prime}}$. If
$Z^{\prime}\neq Y^{\prime}_{\ell^{\prime}}$ then either
$Q^{\prime}_{\ell^{\prime}}=\\{q_{\ell}\\}$ is a point but
$\mu_{\ell}>\mu^{\prime}_{\ell^{\prime}}$ or $Q^{\prime}_{\ell^{\prime}}$ is
not a point.

In the first case, there is a $\beta^{\prime}$-Hölder triangle
$\tilde{T}^{\prime}=T(\tilde{\gamma}^{\prime}_{1},\tilde{\gamma}^{\prime}_{2})$
such that $\beta^{\prime}<\mu_{\ell}$, $V(\tilde{T}^{\prime})\subset
Y^{\prime}_{\ell^{\prime}}$ and $Z^{\prime}\cap V(T^{\prime})$ is a
$\mu_{\ell}$-zone. Let
$\tilde{T}=T(\tilde{\gamma}_{1},\tilde{\gamma}_{2})\subset T$ be a
$\beta^{\prime}$-Hölder triangle, where $\tilde{\gamma}_{1}$ and
$\tilde{\gamma}_{2}$ are two arcs in $T$ such that
$tord(\tilde{\gamma}_{1},\tilde{\gamma}^{\prime}_{1})=tord(\tilde{\gamma}_{2},\tilde{\gamma}^{\prime}_{2})=q_{\ell}$.
Since $Y^{\prime}_{\ell^{\prime}}$ is a pizza slice zone, $\tilde{T}^{\prime}$
is elementary with respect to $g$, and the pair
$(\tilde{T},\tilde{T}^{\prime})$ satisfies (5). It follows from Theorem 3.20
applied to $\tilde{T}^{\prime}$ that $\tilde{T}$ is elementary with respect to
$f$, $Q_{f}(\tilde{T})=\\{q_{\ell}\\}$ and $Z\cap V(\tilde{T})$ is a
$\mu_{\ell}$-zone. Since $\beta^{\prime}<\mu_{\ell}$, this contradicts the
assumption that $Y_{\ell}$ is a minimal pizza slice zone.

The arguments for the second case, when $Q_{\ell}$ is a point but
$Q^{\prime}_{\ell^{\prime}}$ is not a point, are similar: one can find a
Hölder triangle $\tilde{T}^{\prime}\subset T^{\prime}$ such that
$Q_{g}(\tilde{T}^{\prime})=Q^{\prime}_{\ell^{\prime}}$ is not a point,
$\tilde{T}^{\prime}$ is elementary with respect to $g$, and
$V(\tilde{T}^{\prime})\cap Z^{\prime}$ is a $\mu_{\ell}$-zone. Then there is a
Hölder triangle $\tilde{T}\subset T$ such that the pair
$(\tilde{T},\tilde{T}^{\prime})$ satisfies (5). Theorem 3.20 applied to
$\tilde{T}^{\prime}$ implies that $Q_{f}(\tilde{T})$ is not a point, while the
width function of $\tilde{T}$ is affine, a contradiction with the assumption
that $Y_{\ell}$ is a minimal pizza slice zone.

Suppose now that $Q_{\ell}=[q_{\ell-1},q_{\ell}]$ is not a point. Then
Proposition 3.9 and Corollary 3.10 applies to each $q$-order zone
$Z_{q}\subset Y_{\ell}$ for $f$ when
$q\in\dot{Q}_{\ell}=(q_{\ell-1},q_{\ell})$, but may be not applicable when
$q=q_{\ell-1}$ or $q=q_{\ell}$ if $\mu_{\ell}(q)=q$. For $q\in\dot{Q}_{\ell}$,
let $Z^{\prime}_{q}\subset V(T^{\prime})$ be the $q$-order zone for $g$, of
order $\mu_{\ell}(q)$, corresponding to $Z_{q}$. Then
$Z=\bigcup_{q\in\dot{Q}_{\ell}}Z_{q}$ is a pizza slice zone for $f$, and
$Z^{\prime}=\bigcup_{q\in\dot{Q}_{\ell}}Z^{\prime}_{q}$ is a pizza slice zone
for $g$. Let $\subset Y^{\prime}_{\ell^{\prime}}\supset\dot{Z}^{\prime}$ be
the maximal pizza slice zone for a minimal pizza associated with $g$, of order
$\mu^{\prime}_{\ell^{\prime}}$. The same arguments as above show that
$Q^{\prime}_{\ell^{\prime}}=Q_{\ell}$ and
$\mu^{\prime}_{\ell^{\prime}}\equiv\mu_{\ell}$.

Note that the pairs of zones $(Z_{q},Z^{\prime}_{q})$ are either all
positively oriented or all negatively oriented (see Definition 3.12).
Accordingly, either (7) or (8) holds for the pairs of maximal pizza slice
zones $(Y_{\ell},Y^{\prime}_{\ell^{\prime}})$. ∎

######  Definition 4.8.

Let $T=T(\gamma_{1},\gamma_{2})$ and
$T^{\prime}=T(\gamma^{\prime}_{1},\gamma^{\prime}_{2})$ be two normally
embedded Hölder triangles, oriented from $\gamma_{1}$ to $\gamma_{2}$ and from
$\gamma^{\prime}_{1}$ to $\gamma^{\prime}_{2}$ respectively, satisfying
condition (5). Let $\\{T_{\ell}\\}$ and $\\{T^{\prime}_{\ell^{\prime}}\\}$ be
minimal pizzas on $T$ and $T^{\prime}$ for the distance functions
$f(x)=dist(x,T^{\prime})$ and $g(x^{\prime})=dist(x^{\prime},T)$ respectively,
ordered according to the orientations of $T$ and $T^{\prime}$. Then, according
to Proposition 4.7, there is a canonical one-to-one correspondence
$\ell^{\prime}=\tau(\ell)$ between the sets of non-transversal pizza slices
$T_{\ell}$ for a minimal pizza on $T$ associated with
$f(x)=dist(x,T^{\prime})$, ordered according to the orientation of $T$, and
the set of non-transversal pizza slices $T^{\prime}_{\ell^{\prime}}$ for a
minimal pizza on $T^{\prime}$ associated with
$g(x^{\prime})=dist(x^{\prime},T)$, ordered according to the orientation of
$T^{\prime}$. This defines a _characteristic correspondence_ $\tau$ between
the sets of non-transversal pizza slices of $T$ and $T^{\prime}$. In
particular, these two sets have the same number of elements. We say that a
pair of non-transversal pizza slice zones $Y_{\ell}$ and
$Y^{\prime}_{\ell^{\prime}}$ where $\ell^{\prime}=\tau(\ell)$, and a pair of
non-transversal pizza slices $T_{\ell}$ and $T^{\prime}_{\ell^{\prime}}$, is
_positively oriented_ if (7) holds and _negatively oriented_ otherwise (see
Definition 3.16). Thus $\tau$ is a _signed_ correspondence, with the signs $+$
and $-$ assigned to the positively and negatively oriented pairs of non-
transversal pizza slice zones.

######  Remark 4.9.

For each pair $(T_{\ell},T^{\prime}_{\ell^{\prime}})$ of non-transversal pizza
slices, where $\ell^{\prime}=\tau(\ell)$, the signed correspondence $\tau$
defines a correspondence between the two pizza zones $D_{\ell-1}$ and
$D_{\ell}$ and the two pizza zones $D^{\prime}_{\ell^{\prime}-1}$ and
$D^{\prime}_{\ell^{\prime}}$, in the same (resp., opposite) order if the pair
is positively (resp., negatively) oriented. This correspondence between a
subset of pizza zones of $T$ and a subset of pizza zones of $T^{\prime}$ may
be not one-to-one: a pizza zone of $T$ common to two non-transversal pizza
slices may correspond to two different pizza zones of $T^{\prime}$, and two
different pizza zones of $T$ may correspond to the same pizza zone of
$T^{\prime}$ (see Fig. 7). However, it is one-to-one on the set of those pizza
zones which are also maximum zones (see Proposition 4.10).

Figure 7. Two normally embedded Hölder triangles in Remark 4.9. Shaded disks
indicate pizza zones of minimal pizzas on $T$ and $T^{\prime}$. Assuming
$q_{1}=q_{4}>\mu_{1}=\mu_{4}$ and $q_{2}=\mu_{2}=\mu_{3}<q_{3}$, there are
four non-transversal pairs of pizza slices: $(T_{1},T^{\prime}_{1})$,
$(T_{2},T^{\prime}_{3})$, $(T_{3},T^{\prime}_{2})$, $(T_{4},T^{\prime}_{4})$.
The correspondence $\tau(1)$ maps $D_{1}$ to $D^{\prime}_{1}$, while $\tau(2)$
maps $D_{1}$ to $D^{\prime}_{3}$.

######  Proposition 4.10.

Let $T=T(\gamma_{1},\gamma_{2})$ and
$T^{\prime}=T(\gamma^{\prime}_{1},\gamma^{\prime}_{2})$ be normally embedded
Hölder triangles, oriented from $\gamma_{1}$ to $\gamma_{2}$ and from
$\gamma^{\prime}_{1}$ to $\gamma^{\prime}_{2}$ respectively, satisfying
condition (5). Let $\\{T_{\ell}\\}$ and $\\{T^{\prime}_{\ell^{\prime}}\\}$ be
minimal pizzas on $T$ and $T^{\prime}$ for the distance functions
$f(x)=dist(x,T^{\prime})$ and $g(x^{\prime})=dist(x^{\prime},T)$ respectively,
ordered according to the orientations of $T$ and $T^{\prime}$. Let
$(T_{\ell},T^{\prime}_{\ell^{\prime}})$, where $\ell^{\prime}=\tau(\ell)$, be
a pair of non-transversal pizza slices such that one of the pizza zones of
$T_{\ell}$, say $D=D_{\ell}$, is a maximum zone $M_{i}\subset V(T)$. Then the
corresponding pizza zone $D^{\prime}$ of $T^{\prime}_{\ell^{\prime}}$ (either
$D^{\prime}=D^{\prime}_{\ell^{\prime}}$ for a positively oriented pair
$(T_{\ell},T^{\prime}_{\ell^{\prime}})$ or
$D^{\prime}=D^{\prime}_{\ell^{\prime}-1}$ for a negatively oriented pair) is a
maximum zone $M^{\prime}_{i^{\prime}}\subset V(T^{\prime})$, where
$i^{\prime}=\sigma(i)$.

###### Proof.

If $D$ is a boundary arc of $T$ then the statement follows from Proposition
4.4, since $D^{\prime}$ is also a boundary arc of $T^{\prime}$ and a
(singular) maximum zone.

If $D=D_{\ell}$ is not a boundary arc, and both maximal pizza slice zones
$Y_{\ell}$ and $Y_{\ell+1}$ containing $D_{\ell}$ are non-transversal, then
Proposition 4.7 implies that the corresponding zones in $V(T^{\prime})$ are
either $Y^{\prime}_{\ell^{\prime}}$ and $Y^{\prime}_{\ell^{\prime}+1}$ (if
$(Y_{\ell},Y^{\prime}_{\ell^{\prime}})$ is a positively oriented pair) or
$Y^{\prime}_{\ell^{\prime}}$ and $Y^{\prime}_{\ell^{\prime}-1}$ (if
$(Y_{\ell},Y^{\prime}_{\ell^{\prime}})$ is a negatively oriented pair). In
both cases, Proposition 4.7 implies that $D^{\prime}$ is a maximum zone such
that $tord(D,D^{\prime})=tord(D,T^{\prime})=tord(D^{\prime},T)=q_{\ell}$, thus
$i^{\prime}=\sigma(i)$.

Otherwise, if $Y_{\ell}$ is a non-transversal zone but $Y_{\ell+1}$ is
transversal, Lemma 3.18 implies that there are two $\tilde{\beta}$-Hölder
triangles $\tilde{T}=T(\tilde{\gamma}_{1},\tilde{\gamma}_{2})\subset
T_{\ell+1}$ and
$\tilde{T}^{\prime}=T(\tilde{\gamma}^{\prime}_{1},\tilde{\gamma}^{\prime}_{2})\subset
T^{\prime}$ satisfying (5), where $\tilde{\beta}<q_{\ell}$, such that
$\tilde{T}^{\prime}\cap T^{\prime}_{\ell}=\\{\tilde{\gamma}^{\prime}_{1}\\}$,
$\tilde{\gamma}_{1}\in D$, $\tilde{\gamma}^{\prime}_{1}\in D^{\prime}$ and
$tord(\tilde{\gamma}_{1},\tilde{\gamma}^{\prime}_{1})=q_{\ell}$. Theorem 3.20
applied to $\tilde{T}$ and $\tilde{T}^{\prime}$ implies that $D^{\prime}$ is a
maximum zone such that
$tord(D,D^{\prime})=tord(D,T^{\prime})=tord(D^{\prime},T)=q_{\ell}$, thus
$i^{\prime}=\sigma(i)$. ∎

######  Proposition 4.11.

Let $T=T(\gamma_{1},\gamma_{2})$ and
$T^{\prime}=T(\gamma^{\prime}_{1},\gamma^{\prime}_{2})$ be two normally
embedded Hölder triangles, oriented from $\gamma_{1}$ to $\gamma_{2}$ and from
$\gamma^{\prime}_{1}$ to $\gamma^{\prime}_{2}$ respectively, satisfying
condition (5). Then the sign assigned to each pair
$(T_{\ell},T^{\prime}_{\ell^{\prime}})$ of non-transversal pizza slices such
that $\ell^{\prime}=\tau(\ell)$ is completely determined by the minimal pizzas
$\\{T_{\ell}\\}$ and $\\{T^{\prime}_{\ell^{\prime}}\\}$, the characteristic
permutation $\sigma$ and the characteristic correspondence $\tau$.

###### Proof.

Let $Y_{\ell}\subset V(T)$ and $Y^{\prime}_{\ell^{\prime}}\subset
V(T^{\prime})$ be two non-transversal pizza slice zones such that
$\ell^{\prime}=\tau(\ell)$. According to Definition 4.8, the pair
$(Y_{\ell},Y^{\prime}_{\ell^{\prime}})$ is positively oriented if (7) holds
and negatively oriented if (8) holds. If $Q_{\ell}=[q_{\ell},q_{\ell+1}]$ is
not a point then $q_{\ell}\neq q_{\ell+1}$, thus
$Q^{\prime}_{\ell^{\prime}}=[q^{\prime}_{\ell^{\prime}},q^{\prime}_{\ell^{\prime}+1}]$
is also not a point, and the pair is positive when
$q_{\ell}=q^{\prime}_{\ell^{\prime}}$ and negative otherwise. If
$Q_{\ell}=\\{q_{\ell}\\}$ is a point then $\mu_{\ell}<q_{\ell}$, and each of
the pizza zones $D_{\ell}$ and $D_{\ell+1}$ is either a maximum or a minimum
zone. If, say, $D_{\ell}=M_{i}$ is a maximum zone, then the pair is positive
when $D^{\prime}_{\ell^{\prime}}=M_{\sigma(i)}$ and negative otherwise.

The case when each of them contains a boundary arc is trivial, so we may
assume that they are interior zones. If $Q_{\ell}=Q_{\ell^{\prime}}$ is not a
point then the sign is uniquely determined by the maxima of non-constant
affine functions $\mu_{\ell}(q)\equiv\mu^{\prime}_{\ell^{\prime}}(q)$. If
$Q_{\ell}=Q^{\prime}_{\ell^{\prime}}=\\{q_{\ell}\\}>\mu_{\ell}$ is a point,
then the pizza zones $D_{\ell-1}$ and $D_{\ell}$ correspond to the pizza zones
$D^{\prime}_{\ell^{\prime}-1}$ and $D^{\prime}_{\ell^{\prime}}$ in the same
order if the sign is positive, and in the opposite order if the sign is
negative. Note that each of these zones is either a maximum or a minimum zone,
since on one side of each of them $q$ is constant. The correspondence between
the pizza slice zones sends a maximum pizza zone in $V(T)$ to a maximum pizza
zone in $V(T^{\prime})$, and a minimum pizza zone in $V(T)$ to a minimum pizza
zone in $V(T^{\prime})$. If one of the pizza zones in $V(T)$ is a maximum zone
and another is a minimum zone, then the same is true for the pizza zones in
$V(T^{\prime})$, and the correspondence is uniquely defined. If both pizza
zones are maximum zones then the correspondence is defined by $\sigma$. If
both pizza zones in $V(T)$ are minimum zones, since $q_{\ell}>\mu_{\ell}$, the
two maximum pizza zones in $V(T)$ closest to $Y_{\ell}$ are mapped by $\sigma$
to the two maximum pizza zones in $V(T^{\prime})$ closest to
$Y^{\prime}_{\ell^{\prime}}$ in the same order if the sign is positive and in
the opposite order if the sign is negative: one can only get from one side of
$Y_{\ell}$ to another side through a part of $T^{\prime}$ where
$q\leq\mu_{\ell}$. ∎

######  Definition 4.12.

_Let $T=T(\gamma_{1},\gamma_{2})$ and
$T^{\prime}=T(\gamma^{\prime}_{1},\gamma^{\prime}_{2})$ be two normally
embedded Hölder triangles, oriented from $\gamma_{1}$ to $\gamma_{2}$ and from
$\gamma^{\prime}_{1}$ to $\gamma^{\prime}_{2}$ respectively, satisfying
condition (5). Let $\\{T_{\ell}\\}$ and $\\{T^{\prime}_{\ell^{\prime}}\\}$ be
minimal pizzas on $T$ and $T^{\prime}$ for the distance functions
$f(x)=dist(x,T^{\prime})$ and $g(x^{\prime})=dist(x^{\prime},T)$ respectively,
ordered according to the orientations of $T$ and $T^{\prime}$. A
$\sigma\tau$-_pizza_ on $T\cup T^{\prime}$ is a triplet consisting of the pair
of minimal pizzas $\\{T_{\ell}\\}$ and $\\{T^{\prime}_{\ell^{\prime}}\\}$, the
characteristic permutation $\sigma$ of the maximum pizza zones in $V(T)$ and
$V(T^{\prime})$, and the characteristic correspondence $\tau$ of the non-
transversal pizza slices of $T$ and $T^{\prime}$. Two $\sigma\tau$-pizzas
$(\\{T_{\ell}\\},\,\\{T^{\prime}_{\ell^{\prime}}\\},\,\sigma_{T},\,\tau_{T})$
on $T\cup T^{\prime}$ and
$(\\{S_{\ell}\\},\,\\{S^{\prime}_{\ell^{\prime}}\\},\,\sigma_{S},\,\tau_{S})$
on $S\cup S^{\prime}$ are _combinatorially equivalent_ if the pairs
$(\\{T_{\ell}\\},\,\\{T^{\prime}_{\ell^{\prime}}\\})$ and
$(\\{S_{\ell}\\},\,\\{S^{\prime}_{\ell^{\prime}}\\})$ are combinatorially
equivalent, $\sigma_{T}=\sigma_{S}$ and $\tau_{T}=\tau_{S}$._

######  Theorem 4.13.

Let $(T,T^{\prime})$ and $(S,S^{\prime})$ be two oriented pairs of normally
embedded Hölder triangles satisfying condition (5). If there is an
orientation-preserving outer bi-Lipschitz homeomorphism $H:T\cup T^{\prime}\to
S\cup S^{\prime}$ such that $H(T)=S$ and $H(T^{\prime})=S^{\prime}$, then the
$\sigma\tau$-pizzas of the pairs $(T,T^{\prime})$ and $(S,S^{\prime})$ are
combinatorially equivalent.

###### Proof.

Let $f_{T}(x)=dist(x,T^{\prime})$, $g_{T}(x^{\prime})=dist(x^{\prime},T)$,
$f_{S}(y)=dist(y,S^{\prime})$ and $g_{S}(y^{\prime})=dist(y^{\prime},S)$ be
the distance functions defined on $T,\,T^{\prime},\,S$ and $S^{\prime}$
respectively. Let $M_{i}\subset V(T)$ and $M^{\prime}_{i^{\prime}}\subset
V(T^{\prime})$ be the maximum zones for the pair $(T,T^{\prime})$, and let
$N_{i}\subset V(S)$ and $N^{\prime}_{i^{\prime}}\subset V(S^{\prime})$ be the
maximum zones for the pair $(S,S^{\prime})$.

Since $H$ is an outer bi-Lipschitz homeomorphism, $f_{T}$ is Lipschitz contact
equivalent to $f_{S}$, and $g_{T}$ is Lipschitz contact equivalent to $g_{S}$.
Theorem 2.19 implies that the corresponding pairs of minimal pizzas
$(\\{T_{\ell}\\},\,\\{T^{\prime}_{\ell^{\prime}}\\})$ and
$(\\{S_{\ell}\\},\,\\{S^{\prime}_{\ell^{\prime}}\\})$ are combinatorially
equivalent. Accordingly, $H$ maps each maximum zone $M_{i}$ to the maximum
zone $N_{i}$, and each maximum zone $M^{\prime}_{i^{\prime}}$ to the maximum
zone $N^{\prime}_{i^{\prime}}$. A pair of maximum zones
$(M_{i},M^{\prime}_{i^{\prime}})$, where $i^{\prime}=\sigma_{T}(i)$, is mapped
to the pair of maximum zones $(N_{i},N^{\prime}_{i^{\prime}})$ preserving the
order of contact between these zones. This implies that
$i^{\prime}=\sigma_{S}(i)$, thus the permutations $\sigma_{T}$ and
$\sigma_{S}$ are equal.

Moreover, since $H$ preserves the tangency orders between arcs, it maps each
maximal pizza slice zone $Y_{\ell}$ of a minimal pizza on $T$ associated with
$f_{T}$ to the maximal pizza slice zone $Z_{\ell}$ of a minimal pizza on $S$
associated with $f_{S}$, and each maximal pizza slice zone
$Y^{\prime}_{\ell^{\prime}}$ of a minimal pizza on $T^{\prime}$ associated
with $g_{T}$ to the maximal pizza slice zone $Z^{\prime}_{\ell^{\prime}}$ of a
minimal pizza on $S^{\prime}$ associated with $g_{S}$, with the corresponding
width functions preserved. Accordingly, if
$(T_{\ell},T^{\prime}_{\ell^{\prime}})$ is a non-transversal pair of pizza
slices of minimal pizzas associated with $f_{T}$ and $g_{T}$, where
$\ell^{\prime}=\tau_{T}(\ell)$, then
$(H(T_{\ell}),\,H(T^{\prime}_{\ell^{\prime}}))$ is a non-transversal pair of
pizza slices of minimal pizzas associated with $f_{S}$ and $g_{S}$, such that
$V(H(T_{\ell}))\subset Z_{\ell}$ and $V(H(T^{\prime}_{\ell^{\prime}}))\subset
Z^{\prime}_{\ell^{\prime}}$. This implies that $\ell^{\prime}=\tau_{S}(\ell)$,
thus the correspondences $\tau_{T}$ and $\tau_{S}$ are equal. Proposition 4.11
implies that $\tau_{T}$ and $\tau_{S}$ are equal also as signed
correspondences. ∎

The following conjecture states that, conversely, two pairs of normally
embedded Hölder triangles satisfying condition (5) with the same
$\sigma\tau$-pizza invariant are outer bi-Lipschitz equivalent, thus the
$\sigma\tau$-pizza is a complete combinatorial invariant of an outer bi-
Lipschitz equivalence class of pairs of normally embedded Hölder triangles.

######  Conjecture 4.14.

Let $(T,T^{\prime})$ and $(S,S^{\prime})$ be two ordered oriented pairs of
normally embedded Hölder triangles satisfying condition (5). If the
$\sigma\tau$-pizza of the pair $(T,T^{\prime})$ is combinatorially equivalent
to the $\sigma\tau$-pizza of the pair $(S,S^{\prime})$, then there is an
orientation-preserving outer bi-Lipschitz homeomorphism $H:T\cup T^{\prime}\to
S\cup S^{\prime}$ such that $H(T)=S$ and $H(T^{\prime})=S^{\prime}$.

## References

  * [1] L. Birbrair, Local bi-Lipschitz classification of 2-dimensional semialgebraic sets. Houston Journal of Mathematics, N3, vol.25, (1999), pp 453-472.
  * [2] L. Birbrair, T. Mostowski. Normal embeddings of semialgebraic sets. Michigan Math. J., 47 (2000), 125–132
  * [3] L. Birbrair, A. Fernandes, A. Gabrielov, V. Grandjean. Lipschitz contact equivalence of function germs in $\mathbb{R}^{2}$. Annali SNS Pisa, 17 (2017), 81–92, DOI 10.2422/2036-2145.201503$\\_$014
  * [4] L. Birbrair, A. Fernandes, J. Costa, M. Ruas. K-bi-Lipschitz equivalence of real function-germs. Proceedings of the American Mathematical Society, Estados Unidos, v. 135, p. 1089-1095, 2007.
  * [5] L. Birbrair, A. Fernandes, Z. Jelonek. On the extension of bi-Lipschitz mappings. Selecta Mathematica 27, Article number: 15 (2021) DOI : 10.1007/s00029-021-00625-6
  * [6] L. Birbrair and R. Mendes.Arc criterion of normal embedding. In Singularities and foliations. geometry, topology and applications, volume 222 of Springer Proc. Math. Stat., pages 549–553. Springer, Cham, 2018.
  * [7] A. Gabrielov, E. Souza. Lipschitz geometry and combinatorics of abnormal surface germs. Selecta Math., New Series, 28:1 (2022). DOI:10.1007/s00029-021-00716-4
  * [8] K. Kurdyka and P. Orro, Distance géodésique sur un sous-analytique, Rev. Mat. Univ. Complut. Madrid 10 (1997), supplementary, 174–182.
  * [9] T. Mostowski. Lipschitz equisingularity Dissertationes Math. (Rozprawy Mat.), 243 (1985) 46 pp.
  * [10] G. Valette Lipschitz triangulations. Illinois J. Math. 49 (2005), issue 3, 953–979