# Optimal measures for $\boldsymbol{p}$-frame energies on spheres

Dmitriy Bilyk  ,  Alexey Glazyrin  ,  Ryan Matzke  ,  Josiah Park  and
Oleksandr Vlasiuk  School of Mathematics, University of Minnesota,
Minneapolis, MN 55455<EMAIL_ADDRESS>School of Mathematical &
Statistical Sciences, The University of Texas Rio Grande Valley, Brownsville,
TX 78520<EMAIL_ADDRESS>School of Mathematics, University of
Minnesota, Minneapolis, MN 55455<EMAIL_ADDRESS>School of Mathematics,
Georgia Institute of Technology, Atlanta, GA 30332<EMAIL_ADDRESS>Department of Mathematics, Florida State University, Tallahassee, FL 32306
<EMAIL_ADDRESS>
###### Abstract.

We provide new answers about the placement of mass on spheres so as to
minimize energies of pairwise interactions. We find optimal measures for the
$p$-frame energies, i.e. energies with the kernel given by the absolute value
of the inner product raised to a positive power $p$. Application of linear
programming methods in the setting of projective spaces allows for describing
the minimizing measures in full in several cases: we show optimality of tight
designs and of the $600$-cell for several ranges of $p$ in different
dimensions. Our methods apply to a much broader class of potential functions,
those which are absolutely monotonic up to a particular order as functions of
the cosine of the geodesic distance. In addition, a preliminary numerical
study is presented which suggests optimality of several other highly symmetric
configurations and weighted designs in low dimensions. In one case we improve
the best known lower bounds on a minimal sized weighted design in
$\mathbb{CP}^{4}$. All these results point to the discreteness of minimizing
measures for the $p$-frame energy with $p$ not an even integer.

###### Key words and phrases:

Potential energy minimization, spherical codes, spherical designs

###### 2000 Mathematics Subject Classification:

Primary 52A40, 52C17; Secondary 41A05

##  1\. Introduction

An intriguing natural phenomenon is the ubiquitous appearance of certain
symmetric structures and configurations as solutions to optimization problems.
In a number of spaces, highly symmetric configurations of points such as the
vertices of the icosahedron on $\mathbb{S}^{2}$ or the minimal vectors of the
Leech lattice on $\mathbb{S}^{23}$ are optimal codes [Le1]. First papers on
$t$-designs made important connections between symmetry and optimality through
pioneering work on linear programming bounds [DGS]. Some highly symmetric
configurations, in addition to being $t$-designs and optimal codes, are also
minimizers of harmonic energies [A, KY2, KY1, Y1, Y2].

For a finite configuration of points on the sphere,
$\mathcal{C}\subset\mathbb{S}^{d-1}$, the discrete $f$-potential energies are
defined as

(1.1)
$E_{f}(\mathcal{C})=\frac{1}{|\mathcal{C}|^{2}}\sum\limits_{x,y\in\mathcal{C}}f(\langle
x,y\rangle).$

(The diagonal terms should be excluded if the kernel $f$ is singular at $1$,
i.e. when $x=y$.) Universally optimal point configurations, i.e. collections
of points $\mathcal{C}$ minimizing the discrete energies $E_{f}$ among all
point sets of fixed cardinality $|\mathcal{C}|$, for all absolutely monotonic
functions $f$ on $[-1,1)$, have been discovered through the linear programming
approach of Cohn and Kumar in [CK].

In contrast to the above setting, in the present paper, rather than
considering configurations of fixed cardinality, we focus on the problem of
minimizing energies over all Borel probability measures, discovering that
surprisingly in many situations the minimizing measures are discrete. For a
kernel function $f\in C[-1,1]$ and a Borel measure $\mu$ on
$\mathbb{S}^{d-1}$, we define the energy integral as

(1.2) $I_{f}(\mu)=\int_{\mathbb{S}^{d-1}}\int_{\mathbb{S}^{d-1}}f(\langle
x,y\rangle)d\mu(x)d\mu(y).$

One is naturally interested in minimizing these energies over
$\mu\in\mathcal{P}(\mathbb{S}^{d-1})$, the set of all Borel probability
measures on $\mathbb{S}^{d-1}$, i.e. finding the equilibrium distribution of
unit mass under the interaction given by the potential function $f$. This
definition is compatible with the discrete energy 1.1 in the sense that

(1.3)
$E_{f}(\mathcal{C})=I_{f}\Big{(}\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\delta_{x}\Big{)},$

and we shall repeatedly abuse the notation when saying that a configuration
$\mathcal{C}$ minimizes the energy $I_{f}(\mu)$, to mean that the
corresponding measure in the right hand side of the above equation minimizes.

While many classical examples, such as the Riesz energy, feature increasing
kernels $f$ which give rise to energies with repulsive interactions (i.e. $f$
is largest when $x=y$ and smallest when $x$ and $y$ are antipodal), we will
concentrate on the attractive-repulsive potentials, which decrease as
functions of geodesic distance, but increase eventually as a function of the
distance: in other words, a pair of points will repel when close together, but
attract when far apart. Such potentials in $\mathbb{R}^{d}$ appear naturally
for self-assembly models in computational chemistry, emerging collective
behavior in population biology, and in many other scientific models [WS, BCL+,
CFP, VUK+, KSU+, CMV, MEB+].

We will mostly consider attractive-repulsive potentials on the sphere which
are symmetric and orthogonalizing, so that $f(t)=f(|t|)$, $f(t)$ is increasing
for $t\in[0,1]$, and $f$ takes its minimal value at zero. For such potentials,
the discrete energy for up to $d$ particles is minimized by collections of
orthogonal vectors. Since in this setting the energy does not change by
replacing any $x$ with $\lambda x$, where $|\lambda|=1$, its analysis
naturally lends itself to the projective space $\mathbb{R}\mathbb{P}^{d-1}$,
where the potential becomes repulsive, and we adopt this approach in the
technical parts of the paper.

The main examples of the above potentials, which motivate the current paper,
are of the form $f(t)=|t|^{p}$, $p>0$, which yield the $p$-frame energies:

(1.4)
$I_{f}(\mu)=\int_{\mathbb{S}_{\mathbb{F}}^{d-1}}\int_{\mathbb{S}_{\mathbb{F}}^{d-1}}|\langle
x,y\rangle|^{p}d\mu(x)d\mu(y),$

where $\mathbb{S}_{\mathbb{F}}^{d-1}=\\{x\in\mathbb{F}^{d}\ |\ \|x\|=1\\}.$
For $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$ this type of energy has a rich
history.

When $p=2$ and $\mathbb{F}=\mathbb{R}$, the discrete version of this energy,
known simply as the frame energy or frame potential, has been introduced by
Benedetto and Fickus [BeF]: they showed that global (as well as local)
minimizers of this energy are precisely unit norm tight frames. These
configurations, which explain the nomenclature “frame energy”, play an
important role in signal processing and other branches of applied mathematics
and behave like overcomplete orthonormal bases. A finite collection of vectors
$\mathcal{C}\subset\mathbb{F}^{d}$ is a tight frame, if for any
$x\in\mathbb{F}^{d}$, and some constant $A>0$, one has an analog of Parseval’s
identity holding for $\mathcal{C}$,

(1.5) $\sum_{y\in\mathcal{C}}\ |\langle x,y\rangle|^{2}\ =A\|x\|^{2}.$

These objects also minimize the continuous energy $I_{f}(\mu)$ for $p=2$, but
there are also other minimizers, such as the surface area, or Haar measure
$\sigma$ on $\mathbb{S}_{\mathbb{F}}^{d-1}$, and, more generally, isotropic
probability measures on the sphere, i.e. those measures for which

$\displaystyle{\int_{\mathbb{S}_{\mathbb{F}}^{d-1}}|\langle
x,y\rangle|^{2}d\mu(y)=\frac{1}{d}},$

holds for all $x\in{\mathbb{S}_{\mathbb{F}}^{d-1}}$.

When $p=4$, this energy plays an important role in connection to complex
maximal equiangular tight frames, also known as symmetric, informationally
complete, positive operator-valued measures (SIC-POVMs), i.e. unit norm tight
frames in $\mathbb{C}^{d}$ which satisfy $|\langle
x,y\rangle|=\operatorname{const}$ for $x\neq y\in\mathcal{C}$ and
$|\mathcal{C}|=d^{2}$ [RBSC]. The existence of these objects is the subject of
Zauner’s conjecture, and much of the numerical evidence for this conjecture
comes from the observation that they minimize the $4$-frame energy among other
energies, as projective $2$-designs, see e.g. [SG] (in what follows, we
demonstrate that they also minimize the $p$-frame energy for $2\leq p\leq 4$).

More generally, for even integers $p$, these energies were considered in
earlier works [Si, We, V], and it is known that for $\mathbb{F}=\mathbb{R}$ or
$\mathbb{C}$ projective $k$-designs are precisely the finite configurations
which minimize the $p=2k$ energy. So, unit norm tight frames are equivalent to
projective $1$-designs, see Section 2.3 for precise definitions. Spherical
$2$-designs are examples of unit norm tight frames, being exactly those which
are center symmetric. These were constructively shown to exist for $d\geq 2$
precisely when the number of points, $N$, satisfies $N\geq d+1$ and $N\neq
d+2$ when $d$ is odd [Mi]. The last restriction does not apply to unit norm
tight frames, and these exist for all $N\geq d$ [BeF]. Surface measure is also
known to be a minimizer for $p\in 2\mathbb{N}$: this can be seen either from
the definition of $k$-designs, or from the fact that the function $f$ is
positive definite in this case (see Proposition 2.3), and was originally
proved in the real case in [Si].

For $p$ not an even integer, optimal distributions of mass for $p$-frame
energies are much less studied, to the point of there only being one result on
these minimizing measures readily found in the literature. It states that
distributing mass equally on the orthoplex or cross-polytope, an orthonormal
basis and its antipodes, gives the unique symmetric minimizer, up to
orthogonal transformations, for any energy with $p\in(0,2)$ [EO].

This result (contained in our Theorem 1.1 below as a special case) points to
an interesting distinction. When $p$ is even, the $p$-frame energy has a
multitude of both continuous, e.g. $\sigma$, and discrete minimizers. However,
this is not the case when $p$ is not an even integer: $\sigma$ is no longer a
minimizer, since the function $f(t)=|t|^{p}$ is not positive definite, and so
the above result, along with our numerical studies, points to existence of
discrete minimizers only.

In this paper we give a first description of minimizers for several dimensions
and some ranges of $p$. The description relies on the notion of tight designs:
designs of high strength, but with few distinct pairwise distances, see
Definition 2.5. We show that if there exists a tight projective $t$-design
(which in the real case is equivalent to a tight spherical $(2t+1)$-design),
then it minimizes the $p$-frame energy for $p\in(2t-2,2t)$. The $600$-cell,
despite not being a tight design, minimizes the $p$-frame energy for
$p\in(8,10)$ among probability measures on $\mathbb{S}^{3}$, as we show in
Section 4.

######  Theorem 1.1.

Let $f(t)=|t|^{p}$, $t\in[-1,1]$.

  1. (i)

If there exists a tight spherical $(2t+1)$-design
$\mathcal{C}\subset\mathbb{S}^{d-1}$, then the measure

$\mu=\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\delta_{x}$

is a minimizer of the $p$-frame energy $I_{f}$ with $2t-2\leq p\leq 2t$ over
$\mu\in\mathcal{P}(\mathbb{S}^{d-1})$.

  2. (ii)

Let $\mathbb{F}=\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$. Assume that there
exists a tight projective $t$-design
$\widetilde{\mathcal{C}}\subset{\mathbb{F}\mathbb{P}}^{d-1}$, and let the code
$\mathcal{C}\subset\mathbb{S}_{\mathbb{F}}^{d-1}$ consist of the representers
of $\widetilde{\mathcal{C}}$ in $\mathbb{S}_{\mathbb{F}}^{d-1}$ according to
(2.1). Then the measure

$\mu=\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\delta_{x}$

is a minimizer of the $p$-frame energy $I_{f}$ with $2t-2\leq p\leq 2t$ over
$\mu\in\mathcal{P}(\mathbb{S}^{d-1}_{\mathbb{F}})$.

  3. (iii)

Let $\mathcal{C}\subset\mathbb{S}^{3}$ denote the $600$-cell. Then the measure

$\mu=\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\delta_{x}$

is a minimizer of the $p$-frame energy $I_{f}$ with $8\leq p\leq 10$ over
$\mu\in\mathcal{P}(\mathbb{S}^{3})$.

For parts (i)-(ii) of the above theorem we also prove a uniqueness statement:
more precisely, whenever the corresponding statements hold, and additionally
$p$ is not an endpoint of the interval, i.e. $p\in(2t-2,2t)$, all minimizers
have to be tight designs (although not necessarily coinciding with
$\mathcal{C}$), in particular, they have to be discrete. See Section 3.5 for
more details.

We observe that part (i) is essentially contained in part (ii) with
$\mathbb{F}=\mathbb{R}$: indeed, odd-strength tight spherical designs are
necessarily symmetric, and by taking one point in each antipodal pair one
obtains a tight projective design (see Sections 2.3–2.4 for a more extensive
discussion).

Minimizing the continuous energy (1.4) over all measures and obtaining
discrete minimizers allows us to make new conclusions about the minimizing
configurations of the discrete energies (1.1) for certain values of the
cardinality $N$. One directly obtains the following corollary:

######  Corollary 1.2.

Let $\mathbb{F}$, $d$, $p$, and $\mathcal{C}$ be as in any of the parts of
Theorem 1.1, and let $N=k|\mathcal{C}|$, $k\in\mathbb{N}$. Then $N$-point
discrete $p$-frame energy is minimized by the configuration $\mathcal{C}$
repeated $k$ times, i.e.

(1.6)
$\min_{\begin{subarray}{c}\mathcal{C}^{\prime}\subset\mathbb{S}^{d-1}_{\mathbb{F}}\\\
|\mathcal{C}^{\prime}|=N\end{subarray}}\frac{1}{N^{2}}\sum\limits_{x,y\in\mathcal{C^{\prime}}}\big{|}\langle
x,y\rangle\big{|}^{p}=I_{|t|^{p}}\Big{(}\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\delta_{x}\Big{)}.$

Thus, for example, if $N$ is a multiple of $6$, then repeated copies of a
“half” of the icosahedron minimize the $N$-point $p$-frame energy on
$\mathbb{S}^{2}$ for $p\in[2,4]$.

The arguments proving Theorem 1.1 are strongly reminiscent of those appearing
in [CK] and are based on the linear programming method which goes back to
Delsarte and Yudin [De, Y1]. Theorem 1.1 is a consequence of a much more
general Theorem 3.7. The latter theorem, in fact, demonstrates that tight
$t$-designs possess a certain universality property: they minimize the energy
for all strictly monotonic functions of degree exactly $t$ over all
probability measures, see Section 3 for details.

The proof of optimality for the $600$-cell is computer assisted and makes use
of the fact that the averages of spherical harmonics over the $600$-cell
vanish for a few orders above its maximal degree as a spherical design – the
same idea was used in the proof of universal optimality of the $600$-cell in
[CK], as well as earlier in [A, A1]. This allows us to construct a collection
of interpolating polynomials $h$ for each $p$ which have the desired
properties of lying below $f$, agreeing with $f$ on the distances appearing in
$\mathcal{C}$, and finally being positive definite, the last of which is
checked using interval arithmetic. The details of the proof are taken up in
Section 4.

We collect all the necessary preliminary material in Section 2: Section 2.1
contains the discussion of relevant properties of compact 2-point homogeneous
connected spaces; Section 2.2 explains the specifics of minimizing energy
functionals over probability measures on such spaces; Section 2.3 introduces
designs, and, in particular, tight designs; and Section 2.4 describes the
transference between energies on projective spaces and spheres, which connects
Theorem 3.7 to Theorem 1.1.

In our study extensive numerical experiments were conducted. The results of
these experiments are collected in Table 1 for the real case, and Table 2 for
the complex case. Unlike the case of tight designs, optimal weights for these
configurations are generally not equal and thus must be computed for each
relevant value of $p$. Each table gives the minimal support size of a
conjectured optimal point set: when a configuration on the sphere is origin-
symmetric, this minimal support size equals half of the size of the named
configuration. For example, the icosahedron has twelve vertices, however $6$
vertices on one hemisphere suffice to give a minimizer of the $3$-frame energy
on $\mathbb{S}^{2}$. We give additional details in Section 5 for these
conjectured minimizers of the $p$-frame energies. More details on symmetry of
measures and relations between spheres and projective spaces may be found in
Section 2.4.

Our experimental results together with Theorem 1.1 lead us to believe that
clustering of minimizers is a general phenomenon when $p$ is not an even
integer.

######  Conjecture 1.3.

In all dimensions $d\geq 2$ and for all $p>0$ such that $p\not\in
2\mathbb{N}$, the minimizing measures of the $p$-frame energy (1.4) are
discrete.

This conjecture is additionally supported by the fact that discreteness of
minimizers is known for certain attractive-repulsive potentials on
$\mathbb{R}^{d}$ [CFP] and has been conjectured for some other potentials on
the sphere, e.g. those appearing in [FS], see also Section 8.2. While we still
have yet to establish Conjecture 1.3 and prove discreteness, in our companion
paper [BGM+] we show that on $\mathbb{S}^{d-1}$, whenever $p$ is not even, the
support of the measure minimizing the $p$-frame potential necessarily has
empty interior.

In addition to the conjectured discreteness of minimizers our initial study
gave rise to surprisingly symmetric minimizers for $p$-frame energies,
suggesting that further investigation might give new interesting spherical
codes. While nearly all of the minimizing configurations arising from our
numerical experiments have appeared before in the coding theory literature, we
did however discover a new code in $\mathbb{C}^{5}$ of $85$ vectors which in
turn gives a new bound for a minimal sized weighted projective $3$-design. We
detail a construction of this code and its properties in Section 5.1.

Section 6 extends some of our results to non-compact settings. In Section 7 we
apply the results of Theorem 1.1 to the problems of minimizing mixed volumes
of convex bodies, and in Section 8 we apply the methods of linear programming,
similar to those employed in Theorems 1.1 and 3.7, to the optimization of
other kernels, motivated in part by questions from mathematical physics, see
[FS].

Table 1. Optimal and conjectured optimal configurations for $p$-frame energies on $\mathbb{RP}^{d-1}$. Energies are evaluated in most cases at the odd integer which is the midpoint of the interval given. The range $q-$ configurations are obtained as limiting configurations as $p$ tends to $q$ from below. For these configurations, the energy is evaluated for the even limit value. Among the configurations which are not tight, the $600$-cell is the only configuration which is proved to be optimal. $d$ | $N$ | Energy | Range of $p$ | Tight | Name
---|---|---|---|---|---
$2$ | $N$ | $(*)$ | $[2N-4,2N-2]$ | $t$ | regular $2N$-gon
$d$ | $d$ | $1/d$ | $[0,2]$ | $t$ | orthonormal basis
$3$ | $6$ | $0.241202265916660$ | $[2,4]$ | $t$ | icosahedron
$3$ | $11$ | $0.142857142857143$ | $6-$ |  | Reznick design
$3$ | $16$ | $0.124867143799450$ | $[6,8]$ |  | icosahedron and dodecahedron
$4$ | $11$ | $0.125000000000000$ | $4-$ |  | small weighted design
$4$ | $24$ | $0.096277507157493$ | $[4,6]$ |  |  $D_{4}$ root vectors
$4$ | $60$ | $0.047015486159502$ | $[8,10]$ |  |  $600$-cell
$5$ | $16$ | $0.118257675970387$ | $[2,4]$ |  | hemicube
$5$ | $41$ | $0.061838820473855$ | $[4,6]$ |  | Stroud design
$6$ | $22$ | $0.090559619406078$ | $[2,4]$ |  | cross-polytope and hemicube
$6$ | $63$ | $0.042488105634495$ | $[4,6]$ |  |  $E_{6}$ and $E_{6}^{*}$ roots
$7$ | $28$ | $0.071428571428571$ | $[2,4]$ | $t$ | kissing $E_{8}$
$7$ | $91$ | $0.030645893660944$ | $[4,6]$ |  |  $E_{7}$ and $E_{7}^{*}$ roots
$8$ | $36$ | $0.059098639455782$ | $3$ |  | mid-edges of regular simplex
$8$ | $120$ | $0.022916666666667$ | $[4,6]$ | $t$ |  $E_{8}$ roots
$23$ | $276$ | $0.011594202898551$ | $[2,4]$ | $t$ | equiangular lines
$23$ | $2300$ | $0.002028985507246$ | $[4,6]$ | $t$ | kissing Leech lattice
$24$ | $98280$ | $0.000103419439357$ | $[8,10]$ | $t$ | Leech lattice roots
Table 2. Optimal and conjectured optimal configurations for $p$-frame energies on $\mathbb{CP}^{d-1}$. The energies are evaluated at odd integers. $d$ | $N$ | Energy | Range of $p$ | Tight | Name
---|---|---|---|---|---
$d$ | $d$ | $1/d$ | $[0,2]$ | $t$ | orthonormal basis
$3$ | $9$ | $0.222222222222222$ | $[2,4]$ | $t$ | SIC-POVM
$3$ | $21$ | $0.012610934678518$ | $[4,6]$ |  | union equiangular lines
$4$ | $16$ | $0.146352549156242$ | $[2,4]$ | $t$ | SIC-POVM
$4$ | $40$ | $0.068301270189222$ | $[4,6]$ | $t$ | Eisenstein structure on $E_{8}$
$5$ | $25$ | $0.105319726474218$ | $[2,4]$ | $t$ | SIC-POVM
$5$ | $85$ | $0.041997097378053$ | $[4,6]$ |  |  $O_{10}$ and $W(K_{5})$ roots
$6$ | $36$ | $0.080272843473504$ | $[2,4]$ | $t$ | SIC-POVM
$6$ | $126$ | $0.027777777777778$ | $[4,6]$ | $t$ | Eisenstein structure on $K_{12}$
$d$ | $d^{2}$ | $\frac{1+(d^{2}-1)(1/(d+1))^{3/2}}{d^{2}}$ | $[2,4]$ | $t$ | SIC-POVM (conjectured)

We would like to point out that in many papers, the term $p$-frame potential
is usually used to denote the $p$-frame energy (1.4) or its discrete
counterpart. We find the term “energy” to be more appropriate in this context
and reserve the term “potential” for the kernel $f(t)$ of the energy $I_{f}$.

##  2\. Geometry and functions on 2-point homogeneous spaces

###  2.1. Two-point homogeneous spaces

For convenience, the above discussion mostly assumed the underlying space to
be the unit sphere $\mathbb{S}^{d-1}$. This will no longer be the case, as our
study concerns energy minimization on a broader class of spaces. A metric
space $(\Omega,d)$ is said to be two-point homogeneous, if for every two pairs
of points $x_{1},x_{2}$ and $y_{1},y_{2}$ such that
$d(x_{1},x_{2})=d(y_{1},y_{2})$ there exists an isometry of $\Omega$, mapping
$x_{i}$ to $y_{i},$ $i=1,2$. It is known [W] that any such compact connected
space is either a real sphere $\mathbb{S}^{d-1}$, a real projective space
$\mathbb{RP}^{d-1}$, a complex projective space $\mathbb{CP}^{d-1}$, a
quaternionic projective space $\mathbb{HP}^{d-1}$, or the Cayley projective
plane $\mathbb{OP}^{2}$. Note that it suffices to consider $\mathbb{FP}^{d-1}$
for $d>2$ only, as $\mathbb{FP}^{1}$ is just
$\mathbb{S}^{\operatorname{dim}_{\mathbb{R}}\mathbb{F}}$ [B, p. 170], and so
will not be separately considered in what follows.

Below, $\Omega$ always refers to a compact connected $2$-point homogeneous
space, equipped with the geodesic distance $\vartheta$, normalized to take
values in $[0,\pi]$. We let $\sigma$ denote the unique probability measure
invariant under the isometries of $\Omega$.

The first three types of projective spaces
$\\{\mathbb{FP}^{d-1}:\mathbb{F}=\mathbb{R},\mathbb{C},\mathbb{H}\\}$ have a
simple description: they may be represented as the spaces of lines passing
through the origin in $\mathbb{F}^{d}$,

(2.1) $x\mathbb{F}=\\{x\lambda\ |\ \lambda\in\mathbb{F}\setminus\\{0\\}\\}.$

Observe that the isometry groups $O(d)$, $U(d)$, $Sp(d)$ of the corresponding
vector spaces $\mathbb{F}^{d}$ act transitively on each space, and that the
stabilizers of a line represented by $x\in\mathbb{F}^{d}$ are $O(d-1)\times
O(1)$, $U(d-1)\times U(1)$, and $Sp(d-1)\times Sp(1)$, respectively. Thus one
has [Wo, p. 28] the following quotient representations:

$\displaystyle\mathbb{RP}^{d-1}$ $\displaystyle=O(d)/O(d-1)\times O(1),$
$\displaystyle\mathbb{CP}^{d-1}$ $\displaystyle=U(d)/U(d-1)\times U(1),$
$\displaystyle\mathbb{HP}^{d-1}$ $\displaystyle=Sp(d)/Sp(d-1)\times Sp(1),$

where we write $O(d)$, $U(d)$, $Sp(d)$ for the groups of matrices $X$ over the
respective algebra, satisfying $XX^{*}=I$.

Using the identification (2.1), one can associate each element of
$\mathbb{FP}^{d-1}$ ($\mathbb{F}=\mathbb{R},\mathbb{C},\mathbb{H}$) with a
unit vector $x\in\mathbb{F}^{d}$, $\|x\|=1$, and we shall often abuse notation
by doing so. This gives, in addition to the Riemannian metric $\vartheta$,
another metric, the chordal distance between points $x,y\in\Omega$, defined by

$\rho(x,y)=\sqrt{1-|\langle x,y\rangle|^{2}},$

where $\displaystyle{\langle
x,y\rangle=\sum\limits_{i=1}^{d}{x}_{i}\overline{y}_{i}}$ is the standard
inner product in $\mathbb{F}^{d}$. The chordal distance $\rho(x,y)$ is related
to the geodesic distance $\vartheta(x,y)$ by the equation

$\cos\vartheta(x,y)=1-2\rho(x,y)^{2}=2|\langle x,y\rangle|^{2}-1.$

Since the algebra of octonions is not associative, the line model of (2.1)
fails, and instead a model given by Freudenthal [Fr] is used to describe
$\mathbb{OP}^{d-1}$. It is known [B] that only two octonionic spaces exist:
$\mathbb{OP}^{1}$ and $\mathbb{OP}^{2}$, however $\mathbb{OP}^{1}$ is just
$\mathbb{S}^{8}$, as noted above.

$\mathbb{OP}^{2}$ can be described as the subset of $3\times 3$ Hermitian
matrices $\Pi$ over $\mathbb{O}$, satisfying $\Pi^{2}=\Pi$ and $\text{Tr}\
\Pi=1$ [Sk]. A metric for $\mathbb{OP}^{2}$ is then given by the Frobenius
product,

$\rho(\Pi_{1},\Pi_{2})=\frac{1}{\sqrt{2}}\|\Pi_{1}-\Pi_{2}\|_{F}=\sqrt{1-\langle\Pi_{1},\Pi_{2}\rangle},$

where $\langle\Pi_{1},\Pi_{2}\rangle=\text{Re}\ \text{Tr}\
\frac{1}{2}(\Pi_{1}\Pi_{2}+\Pi_{2}\Pi_{1})$. This is the chordal distance on
$\mathbb{OP}^{2}$ whereas the geodesic distance can be defined through
$\sin{\frac{\vartheta(x,y)}{2}}=\rho(x,y)$, as in the above projective spaces.
All $\Pi$ given as above may be written in the form

$\begin{pmatrix}|a|^{2}&a\overline{b}&a\overline{c}\\\
b\overline{a}&|b|^{2}&b\overline{c}\\\
c\overline{a}&c\overline{b}&|c|^{2}\end{pmatrix},$

where $|a|^{2}+|b|^{2}+|c|^{2}=1$ and $(ab)c=a(bc)$. This gives a
representation of $\mathbb{OP}^{2}$ as the quotient $F_{4}/{\rm Spin}(9)$ [B,
p. 189].

One feature of spaces $\Omega$ that allows for the application of linear
programming methods is the existence of a decomposition of
$L^{2}(\Omega,\sigma)$, the space of complex-valued square-integrable
functions on $\Omega$, into irreducible representations:

$L^{2}(\Omega,\sigma)=\bigoplus\limits_{n\geq 0}V_{n},$

where spaces $V_{n}$ are finite-dimensional and invariant under the isometries
of $\Omega$ (see [Le1]). Moreover, they can be chosen as the eigenspaces of
the Laplace–Beltrami operator on $\Omega$ corresponding to the $n$-th
eigenvalue in the increasing order. Let $Y_{n,k}$,
$k=1,\ldots,\operatorname{dim}V_{n}$, be an orthonormal basis in $V_{n}$.
Because of the invariance of $V_{n}$ and due to the two-point homogeneity of
$\Omega$, the reproducing kernel for $V_{n}$ only depends on the distance
$\vartheta(x,y)$ between points [V]. Furthermore, as a function of

$\tau(x,y):=\cos\vartheta(x,y).$

the reproducing kernel is a polynomial $C_{n}$ of degree $n$, which satisfies

(2.2)
$C_{n}(\tau(x,y))=\frac{1}{\operatorname{dim}V_{n}}\sum\limits_{k=1}^{\operatorname{dim}V_{n}}Y_{n,k}(x)\overline{Y_{n,k}(y)}.$

Formula (2.2) is known as the addition formula, and shows that functions
$C_{n}$ are positive definite on $\Omega$, that is,

$\sum\limits_{1\leq i,j\leq
k}c_{i}\overline{c}_{j}C_{n}(\tau{(x_{i},x_{j})})\geq 0$

for all coefficients $\\{c_{i}\\}_{i=1}^{k}\subset\mathbb{F}$, and all vectors
$\\{x_{i}\\}_{i=1}^{k}\subset\Omega$.

The polynomials $C_{n}$ given by (2.2) satisfy $C_{n}(1)=1$ and are orthogonal
with respect to the probability measure

$d\nu^{(\alpha,\beta)}=\frac{1}{\gamma_{\alpha,\beta}}(1-t)^{\alpha}(1+t)^{\beta}dt,$

where $\alpha=(d-1)\operatorname{dim}_{\mathbb{R}}(\mathbb{F})/{2}-1$ and

(2.3) $\beta=\begin{cases}\alpha,&\text{ if }\Omega=\mathbb{S}^{d-1};\\\
{\operatorname{dim}_{\mathbb{R}}(\mathbb{F})}/{2}-1,&\text{ if
}\Omega=\mathbb{FP}^{d-1},\end{cases}$

and the normalization factor is given by

$\gamma_{\alpha,\beta}=2^{\alpha+\beta+1}B(\alpha+1,\beta+1),$

where $B$ is the beta function. Jacobi polynomials form an orthogonal basis in
$L^{2}([-1,1],d\nu^{(\alpha,\beta)})$; equivalently, the span of
$C_{n}(\tau(x,y))$, $n\geq 0$, is dense in the subset of
$L^{2}(\Omega\times\Omega,\sigma\otimes\sigma)$ consisting of functions that
depend only on the distance between $x$ and $y$.

This allows expanding functions from $L^{2}([-1,1],d\nu^{(\alpha,\beta)})$ in
terms of $C_{n}$:

$f(t)=\sum\limits_{i=0}^{\infty}\widehat{f}_{n}C_{n}(t),\quad\text{where}\quad\widehat{f}_{n}={\operatorname{dim}V_{n}}\int\limits_{-1}^{1}f(t)C_{n}(t)\,d\nu^{(\alpha,\beta)}(t).$

As we have already done above, for a fixed space $\Omega$ we will not indicate
the dependence of polynomials $C_{n}=C_{n}^{(\alpha,\beta)}$ on the indices
$\alpha$, $\beta$. We refer to $\widehat{f}_{n}$ as the Jacobi coefficients of
the function $f$; the normalization $C_{n}(1)=1$ used here is common in the
coding theory community [Sz, Le1].

###  2.2. Energies on 2-point homogeneous spaces

For the space of probability measures $\mathcal{P}(\Omega)$ supported on
$\Omega$, and for a lower semi-continuous function
$f:[-1,1]\rightarrow\mathbb{R}\cup\infty$, the $f$-energy integral is defined
as the functional mapping $\mu$ to

$I_{f}(\mu)=\int_{\Omega}\int_{\Omega}f(\tau(x,y))d\mu(x)d\mu(y).$

Observe that when $\Omega=\mathbb{S}^{d-1}$, we have
$\tau(x,y)=\cos\vartheta(x,y)=\langle x,y\rangle$ and the definition above
coincides with (1.2).

We start by introducing the notion of positive definite functions, which plays
an important role in energy minimization and for the linear programming bounds
we derive later. Below $C[-1,1]=C_{\mathbb{R}}[-1,1]$ denotes the space of
continuous real valued functions on the interval $[-1,1]$.

######  Definition 2.1.

Let $f\in C[-1,1]$. We say that $f$ is positive definite on $\Omega$ if for
any set $\\{x_{1},\ldots,x_{N}\\}\subset\Omega$ the matrix
$\big{[}f\big{(}\tau(x_{i},x_{j})\big{)}\big{]}_{i,j=1}^{N}$ is positive
semidefinite, i.e. for every collection
$\\{c_{1},\ldots,c_{k}\\}\subset\mathbb{C}$ we have

$\sum_{1\leq i,j\leq N}f(\tau(x_{i},x_{j}))c_{i}\overline{c_{j}}\geq 0.$

We have already seen that the Jacobi polynomials $C_{n}$ are positive definite
on $\Omega$, and so their positive linear combinations must also be. It is a
classical fact that this implication can be reversed:

######  Proposition 2.2.

[Bo, Sc, G] A function $f\in C[-1,1]$ is positive definite on $\Omega$ if and
only if $\widehat{f}_{n}\geq 0$ for all $n\geq 0$.

Next we show that positive definite functions $f$ give rise to $f$-energy
integrals which are minimized over probability measures by the surface, or
Haar measure $\sigma$ on $\Omega$. This result appears in a number of papers,
see for instance [DG, BiD]. We adapt the proof given in [BiD] to our purposes,
choosing to work with the real and imaginary parts of the functions $Y_{n,k}$
defined above. By a slight abuse of notation, we use the same notation for
these functions.

######  Proposition 2.3.

Let $f\in C[-1,1]$, $f(t)=\sum\limits_{n=0}^{\infty}\widehat{f}_{n}C_{n}(t)$,
and $\mu\in\mathcal{P}(\Omega)$. Then, the following are equivalent:

  1. (i)

$\widehat{f}_{n}\geq 0$ for all $n\geq 1$,

  2. (ii)

the surface measure $\sigma$ is a minimizer of $I_{f}(\mu)$.

Moreover, $\sigma$ is the unique minimizer of $I_{f}(\mu)$ if and only if
$\widehat{f}_{n}>0$ for all $n\geq 1$.

To prove this statement we use the following lemma, generalizing the behavior
of Fourier expansions with positive coefficients [G, L] to Jacobi expansions
with the same property.

######  Lemma 2.4.

Let $f\in C[-1,1]$, $f(t)=\sum\limits_{n=0}^{\infty}\widehat{f}_{n}C_{n}(t)$,
and $\widehat{f}_{n}\geq 0$ for all $n\geq 1$. Then the Jacobi expansion of
$f$ converges uniformly and absolutely to $f$ on $[-1,1]$.

###### Proof of Proposition 2.3.

We first show that $\sigma$ is a minimizer of $I_{f}(\mu)$. Assume that
$\widehat{f}_{n}\geq 0$ for all $n\geq 1$. Then by the lemma above, the Fubini
theorem, and the addition formula, we have

$\displaystyle I_{f}(\mu)$
$\displaystyle=\sum\limits_{n=0}^{\infty}\widehat{f}_{n}\int_{\Omega}\int_{\Omega}C_{n}(\tau(x,y))d\mu(x)d\mu(y)$
$\displaystyle=\sum\limits_{n=0}^{\infty}\widehat{f}_{n}\cdot\frac{1}{\operatorname{dim}V_{n}}\sum\limits_{n=1}^{\operatorname{dim}V_{n}}\int_{\Omega}\int_{\Omega}Y_{n,k}(x)Y_{n,k}(y)d\mu(x)d\mu(y)$
$\displaystyle=\widehat{f}_{0}+\sum\limits_{n=1}^{\infty}\frac{1}{\operatorname{dim}V_{n}}\cdot\widehat{f}_{n}b_{n,\mu},$
$\displaystyle\geq\widehat{f}_{0}=I_{f}(\sigma).$

The last inequality holds since
$b_{n,\mu}=\sum\limits_{k=1}^{\operatorname{dim}V_{n}}\left[\int_{\Omega}Y_{n,k}(x)d\mu(x)\right]^{2}\geq
0$. If $\widehat{f}_{n}>0$ for all $n\geq 1$, then equality can be achieved
above only if $\mu$ is orthogonal to all spaces $V_{n}$, which directly
implies that $\mu=\sigma$. If $\widehat{f}_{n}<0$ for some $n\geq 1$, then set
$d\mu(x)=(1+\epsilon Y_{n,1}(x))d\sigma(x)$, where $\epsilon>0$ is
sufficiently small so that $(1+\epsilon Y_{n,1}(x))\geq 0$ on $\Omega$.
Orthogonality and the addition formula (or Funk-Hecke formula) give that for
$Y\in\mathcal{H}_{n},$

$\int_{\Omega}f(\tau(x,y))Y(x)d\sigma(x)=\widehat{f}_{n}Y(y)\ \text{ and }\
\int_{\Omega}Y_{n,1}(x)d\sigma=0.$

Thus,

$\displaystyle I_{f}(\mu)$
$\displaystyle=\int_{\Omega}\int_{\Omega}f(\tau(x,y))(1+\epsilon
Y_{n,1}(x))(1+\epsilon Y_{n,1}(y))d\sigma(x)d\sigma(y)$
$\displaystyle=I_{f}(\sigma)+\epsilon^{2}\widehat{f}_{n}\int_{\Omega}Y^{2}_{n,1}(x)d\sigma(x)<I_{f}(\sigma),$

implying that $\sigma$ is not a minimizer for $I_{f}(\mu)$. If
$\widehat{f}_{n}=0$ for some $n\geq 1$, the same argument shows that
$I_{f}(\mu)=I_{f}(\sigma)$, i.e. $\sigma$ is not the unique minimizer. ∎

The $p$-frame energies correspond to taking $\Omega=\mathbb{FP}^{d-1}$
($\mathbb{F}=\mathbb{R}$, $\mathbb{C}$, or $\mathbb{H}$) and $f$ of the form

(2.4) $f(t)=\bigg{(}\frac{1+t}{2}\bigg{)}^{\frac{p}{2}},$

because in this case, since $\tau(x,y)=\cos\vartheta(x,y)=2|\langle
x,y\rangle|^{2}-1$, we have

$f(\tau(x,y))=f\big{(}2|\langle x,y\rangle|^{2}-1\big{)}=|\langle
x,y\rangle|^{p}.$

We shall now prove that, whenever $p$ is an even integer, these energies are
minimized by the uniform measure on $\Omega$.

When $p=2k$ and $\Omega=\mathbb{FP}^{d-1}$
($\mathbb{F}=\mathbb{R},\mathbb{C},$ or $\mathbb{H}$), we have that
$f(t)=2^{-k}\cdot(1+t)^{k}$ is a polynomial. It is standard to check that this
polynomial is positive definite on $\Omega$: this could be done by checking
that the coefficients in its Jacobi expansion are non-negative, but it would
be perhaps simpler to prove it as follows. Observe that, since
$C_{0}^{(\alpha,\beta)}(t)=1$ and
$C_{1}^{(\alpha,\beta)}(t)=\frac{\alpha-\beta}{2(\alpha+1)}+\frac{\alpha+\beta+2}{2(\alpha+1)}\cdot
t$, we have that

$1+t=\frac{2(\alpha+1)}{(\alpha+\beta+2)}\,C_{1}^{(\alpha,\beta)}(t)\,+\,\frac{2(\beta+1)}{\alpha+\beta+2}\,C_{0}^{(\alpha,\beta)}(t).$

Since
$\alpha+1=\frac{d-1}{2}\cdot\operatorname{dim}_{\mathbb{R}}(\mathbb{F})>0$ and
$\beta+1=\frac{1}{2}\cdot\operatorname{dim}_{\mathbb{R}}(\mathbb{F})>0$, we
see that the function $1+t$ is positive definite on $\Omega$. The well known
Schur’s theorem on Hadamard (elementwise) products of positive semidefinite
matrices implies that if $g$ and $h$ are positive definite on $\Omega$, then
so is their product $gh$, and, in particular, all integer powers $g^{n}$ are
positive definite. Hence, the function $f(t)=2^{-k}\cdot(1+t)^{k}$ is positive
definite on $\Omega$, and therefore $I_{f}(\sigma)$ is minimized by the
uniform surface measure $\sigma$.

The minimal values of the $p=2k$ energy may be expressed in elementary
functions for each $\mathbb{F}$. These constants, $c_{\mathbb{F}}(d,k)$, are
given below

$\displaystyle c_{\mathbb{F}}(d,k)$ $\displaystyle=\frac{1\cdot 3\cdot
5\dots(2k-1)}{d\cdot(d+2)\dots(d+2(k-1))},\ $
$\displaystyle\mathbb{F}=\mathbb{R},$ $\displaystyle c_{\mathbb{F}}(d,k)$
$\displaystyle=1/\binom{d+k-1}{k},\ $ $\displaystyle\mathbb{F}=\mathbb{C},$
$\displaystyle c_{\mathbb{F}}(d,k)$ $\displaystyle=(k+1)/\binom{2d+k-1}{k},\ $
$\displaystyle\mathbb{F}=\mathbb{H}.$

When $p$ is not an even integer, the $p$-frame energies are not positive
definite, due to the appearance of negative terms in the Jacobi polynomial
expansion of $f$, hence $\sigma$ does not minimize the $p$-frame energy for
$p\not\in 2\mathbb{N}$.

###  2.3. Designs

We now treat the topic of designs in the compact connected two-point
homogeneous spaces $\Omega$. A finite set $\mathcal{C}\subset\Omega$ is called
a $t$-design if

(2.5)
$\frac{1}{|\mathcal{C}|}\sum\limits_{x\in\mathcal{C}}p(x)=\int_{\Omega}p(x)\,d\sigma_{\Omega}(x)$

holds for all polynomials $p$ of degree at most $t$. Here
$d\sigma_{\Omega}(x)$ is the Haar (or surface) measure on $\Omega$. A
relaxation of the above identity allows the configuration to be weighted, so
that the equality

(2.6)
$\sum\limits_{x\in\mathcal{C}}\omega_{x}p(x)=\int_{\Omega}p(x)\,d\sigma_{\Omega}(x),$

holds for some weights
$\\{\omega_{x}\\}_{x\in\mathcal{C}}\subset\mathbb{R}_{\geq 0}$, satisfying
$\sum_{x\in\mathcal{C}}\omega_{x}=1$, and all polynomials $p$ of degree at
most $t$. Such weighted formulas are called cubature formulas or weighted
designs. In both of the above equations, it is understood that polynomials $p$
may be given explicitly as complex-valued functions which are polynomials in
coordinates of $\mathbb{F}^{d}$, satisfying additionally $p(\alpha x)=p(x)$,
for $|\alpha|=1$, $\alpha\in\mathbb{F}$, in the projective case.

The strength of a (weighted) design is the maximum value of $t$ for which
identity (2.5) (accordingly, (2.6)) holds. A $t$-design can be equivalently
defined as a configuration $\mathcal{C}\subset\Omega$, for which

$\sum\limits_{x,y\in\mathcal{C}}C_{n}(\tau(x,y))=0\quad\text{ for }1\leq n\leq
t.$

Similarly, $\mathcal{C}$ is a $t$-design in $\Omega$ if and only if it
satisfies

$\sum\limits_{x\in\mathcal{C}}Y(x)=0\ \text{ for}\
Y\in\bigoplus\limits_{n=1}^{t}V_{n}.$

Linear programming bounds imply exact constraints on the size of tight
designs, configurations which, in addition to being $t$-designs, have the
smallest possible number of pairwise distances between their elements, for a
design of strength $t$. The exact definition may be given as follows.

######  Definition 2.5.

A discrete set $\mathcal{C}\subset\Omega$ is called a tight $t$-design if one
of the following conditions is satisfied.

  1. (i)

$\mathcal{C}$ is a design of degree $t=2m-1$ and there are $m$ distances
between its distinct elements, including at least one pair diameter apart;

  2. (ii)

$\mathcal{C}$ is a design of degree $t=2m$ and there are $m$ distances between
its distinct elements.

Tight spherical $2$-designs are precisely regular simplices. For $d\geq 3$ and
$t\geq 4$ there are eight tight spherical designs known. Tight odd-degree
spherical designs must be centrally symmetric [DGS], and by choosing points
from each antipode in an odd tight design one arrives at a real projective
tight design. Six of the eight designs mentioned above are odd degree and
correspond to the first six entries in the Table 3. The (remaining) known
tight spherical $4$-designs are the Schläfli configuration of $27$ points in
$\mathbb{S}^{5}$ and the $275$ point arrangement associated with the
McLaughlin group in $\mathbb{S}^{21}$.

Tight spherical designs with $d\geq 3$ and $t\geq 4$ may only exist for
$t=4,5,$ and $7$ with the one exception of the spherical $11$-design formed by
the Leech lattice roots [BD1, BD2]. Finding tight spherical $5$-designs is the
same problem as finding maximal equiangular tight frames, and it is known that
existence of a tight spherical $5$-design in $\mathbb{S}^{d-1}$ is possible
only for $d$ of the form $d=(2k+1)^{2}-2$ for some integer $k\geq 1$; see
[BMV] for details on how these conditions arise. A direct correspondence with
such spherical designs and regular graphs has long been recognized [Se], and
in connection, it is known that for $d=47$ a tight spherical $5$-design cannot
exist [M]. For projective spaces, it is known that no tight $t$-designs exist
in the complex or quaternionic setting whenever $t\geq 4$ and $d\neq 2$ [BH,
H2, L].

A weaker property of a design is sharpness, which will not play a role here.
The paper [CK] proves that sharp designs, and tight designs in particular, are
minimizers for discrete minimization problems with absolutely monotone
kernels. A similar approach allows us to show that tight designs are optimal
for the continuous $p$-frame energy.

Table 3. A list of all known projective tight designs (besides cross-polytopes, SIC-POVMs, and designs in $\mathbb{FP}^{1}$) along with relevant parameters. Here $M$ denotes the strength of the design, $d$ the dimension of the ambient space $\mathbb{F}^{d}$, and $N$ is the size of the design. $d$ | $N$ | $M$ | Inner Products | $\mathbb{F}$ | Name
---|---|---|---|---|---
$3$ | $6$ | $2$ | $1/\sqrt{5}$ | $\mathbb{R}$ | icosahedron
$7$ | $28$ | $2$ | $1/3$ | $\mathbb{R}$ | kissing configuration for $E_{8}$
$8$ | $120$ | $3$ | $0,1/2$ | $\mathbb{R}$ | roots of $E_{8}$ lattice
$23$ | $276$ | $2$ | $1/5$ | $\mathbb{R}$ | equiangular lines
$23$ | $2300$ | $3$ | $0,1/3$ | $\mathbb{R}$ | kissing configuration for $\Lambda_{24}$
$24$ | $98280$ | $5$ | $0,{1}/{4},{1}/{2}$ | $\mathbb{R}$ | roots of $\Lambda_{24}$, the Leech lattice
$4$ | $40$ | $3$ | $0,{1}/{\sqrt{3}}$ | $\mathbb{C}$ | Eisenstein structure on $E_{8}$
$6$ | $126$ | $3$ | $0,1/2$ | $\mathbb{C}$ | Eisenstein structure on $K_{12}$
$5$ | $165$ | $3$ | $0,1/2$ | $\mathbb{H}$ | quaternionic reflection group
$3$ | $819$ | $5$ | $0,1/2,1/\sqrt{2}$ | $\mathbb{O}$ | generalized hexagon of order $(2,8)$

###  2.4. Antipodal symmetry

We observe that the energy $I_{f}$ on the sphere
$\mathbb{S}^{d-1}_{\mathbb{F}}$ for the kernels $f$ with $f(t)=f(|t|)$ remains
the same after averaging over unit multiples of vectors in the support of
$\mu$. Let $U(\mathbb{F})$ be the set of units in $\mathbb{F}$,
$U(\mathbb{F})=\\{c\in\mathbb{F}\ |\ |c|=1\\}$, and $\eta$ be the uniform
measure on $U(\mathbb{F})$. If one defines, for Borel sets
$B\subset\mathbb{S}_{\mathbb{F}}^{d-1}$,

$\nu(B)=\frac{1}{\eta(U_{\mathbb{F}})}\int_{U_{\mathbb{F}}}\mu(cB)d\eta(c),$

then $I_{f}(\nu)=I_{f}(\mu)$ for potential functions $f$ as above. This is the
primary reason it is natural to consider projective spaces
$\mathbb{F}\mathbb{P}^{d-1}$ as the optimization spaces for $p$-frame
energies, as opposed to the spheres, in the cases when the elements
$x\in\mathbb{F}\mathbb{P}^{d-1}$ may be represented by unit vectors in
$\mathbb{F}^{d}$.

This discussion shows that a minimizing measure on the sphere for
$I_{f}(\mu)$, with $f$ as above, can be taken to be symmetric, and that the
problem of minimizing over symmetric measures on spheres is equivalent to
minimizing energy over projective spaces. In particular, this explains part
(i) of Theorem 1.1, since tight spherical $(2t+1)$-designs are necessarily
symmetric and hence correspond to tight real projective $t$-designs.

As a final discussion point on the form of our energies, note that the kernel
$f(t)=(\frac{1+t}{2})^{\frac{p}{2}}$ has as first negative derivative
$f^{(\lceil p/2\rceil+1)}(t)$, $-1<t<1$. This plays an important role in what
follows and it is precisely functions with this property of alternating
derivative sign (for a large enough index) which our results apply to.

##  3\. Optimality of tight designs for kernels absolutely monotonic to degree
$M$

###  3.1. Linear programming

The main goal of this section is to show that for those dimensions and values
of $t$ for which tight designs exist, they are the global minimizers of the
$p$-frame energies for intervals of $p$ between consecutive even integers. We
will use linear programming bounds to this end.

The linear programming method provides bounds for optima in various
optimization problems, and its use is often aided by computational tools,
where a problem is approximated by a finite-dimensional or discretized
counterpart, then solved with a computer. It is surprising that this simple
method provides optimal bounds often. This technique applies to all the
compact $2$-point homogeneous spaces $\Omega$ described above.

Our application of the method can be summed up in the following lemma, which
is a measure-theoretic counterpart of the linear programming bound of Delsarte
and Yudin [De, Y1].

######  Lemma 3.1.

Let $h\in C[-1,1]$ be a positive-definite function, i.e.
$h(t)=\sum\limits_{n=0}^{\infty}\widehat{h}_{n}C_{n}(t)$ and
$\widehat{h}_{n}\geq 0$ for all $n\geq 0$.

  1. (i)

Assume that $h(t)\leq f(t)$ for all $t\in[-1,1]$, then for any
$\mu\in\mathcal{P}(\Omega)$,

$I_{f}(\mu)\geq\widehat{h}_{0}=I_{h}(\sigma).$

  2. (ii)

Assume further that $h$ is a polynomial of degree $k$ and that there exists a
$k$-design $\mathcal{C}\subset\Omega$ such that $h(t)=f(t)$ for each
$t\in\\{\tau(x,y):\,x,y\in\mathcal{C}\\}$. Then for any
$\mu\in\mathcal{P}(\Omega)$,

$I_{f}(\mu)\geq
I_{f}\Big{(}\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\delta_{x}\Big{)},$

i.e. $I_{f}$ is minimized by the uniform distribution on $\mathcal{C}$.

###### Proof.

For the first part observe that

$I_{f}(\mu)\geq I_{h}(\mu)\geq I_{h}(\sigma)=\widehat{h}_{0},$

where the first inequality follows from the fact that $f\geq h$, while the
second one is due to Proposition 2.3, since $h$ is positive definite.

For the second part, one can continue as follows

$I_{h}(\sigma)=I_{h}\Big{(}\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\delta_{x}\Big{)}=I_{f}\Big{(}\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\delta_{x}\Big{)}.$

The first equality follows from the fact that $\mathcal{C}$ is a $k$-design,
and the second one from the fact that $f$ and $h$ coincide on the set
$\\{\tau(x,y):\,x,y\in\mathcal{C}\\}$. Together with part (i) this proves the
statement in part (ii). ∎

This lemma provides insights in two different ways for how the linear
programming method can be applied.

If a candidate $\mathcal{C}$ is available, one can apply part (ii) of Lemma
3.1 by constructing a polynomial $h\leq f$ as a Hermite interpolant of the
function $f$ at the points of $\\{\tau(x,y):\,x,y\in\mathcal{C}\\}$. This
reasoning, which lies behind the proof of Theorems 3.7 and 1.1, explains the
appearance of tight designs: indeed, the number of elements in the set of
interpolation points (i.e. distinct distances between the points of
$\mathcal{C}$) determines the degree of the interpolant $h$ – hence one wants
a design of high strength, but with few mutual distances.

The same reasoning as above applies to the emergence of sharp designs as
universally optimal sets in [CK], and it also explains why this slightly
weaker notion does not suffice for our purposes: since we are working with
general measures rather than point sets with fixed cardinality, we cannot
avoid interpolating at the point $t=1$, which requires a design of higher
strength. The main technical difficulty in this setting is proving positive
definiteness of the Hermite interpolating polynomial $h$. We take this
approach to Theorem 3.7 and carry out the technicalities in Sections 3.2–3.4.

If a suitable candidate is not available, one can still rely on part (i) of
Lemma 3.1 and attempt to optimize the value of the energy $I_{h}(\sigma)$ over
auxiliary positive definite polynomials $h$, obtaining a lower bound for the
energy over all probability measures. If the degree of an auxiliary function
$h$ is bounded by $D$, we have $D+1$ non-negative variables $\widehat{h}_{i}$,
$0\leq i\leq D$, and infinitely many linear constraints $h(t)\leq f(t)$ for
all $t\in[-1,1]$. In order to get the best possible lower bound, we need to
maximize $\widehat{h}_{0}$ given these linear conditions.

This problem is, generally, intractable as a linear optimization problem.
However, when $f$ is a polynomial, the condition $f(t)-h(t)\geq 0$ for all
$t\in[-1,1]$ may be represented as a finite-size positive semi-definite
constraint on the coefficients $\widehat{h}_{i}$. In particular, the
polynomial inequality may be rewritten as a sum-of-squares optimization
problem (see, for instance, [N]) and thus solved as a semi-definite program.

By using sum-of-squares optimization described above, we obtain lower bounds
on the $p$-frame energies over measures on projective spaces when $p$ is an
odd integer. A table of such bounds for real projective spaces
$\mathbb{RP}^{d-1}$, $3\leq d\leq 24$, and $p=3,5,7$, is shown in Table 8 in
the Appendix. The concrete bounds are computed by a series of steps. For the
first step, we fix the degree $D$ of the auxiliary polynomial and solve the
sum-of-squares problem. The numerical solver outputs a polynomial which is
feasible up to a small tolerance. By rounding coefficients, it is then
possible to obtain polynomials which are less than $f$ and positive definite.

Since the choice of the maximal degree $D$ is arbitrary, not much is lost by
rounding, and our bounds in the appendix are thus rounded down to four
significant figures. The last condition $f-h\geq 0$ can be checked using
interval arithmetic, or by hand. We include the coefficients of the auxiliary
polynomials in the supplementary files of the arXiv version of this paper. The
polynomials used for $p=3$ and $p=5$ are of degree $D=6$, while the
polynomials for $p=7$ are of degree $D=8$.

It is interesting to compare the values of conjectured energy minimizers with
the lower bounds obtained using the approach above. We make comparison of
these bounds in Table 4 below for all conjectured optimizers from Tables 1 and
2: observe that the values are indeed close, which motivates our conjectures
about the minimizers. Tight designs are excluded from this table since for
them the lower and the upper bounds coincide as we will show below in Theorem
3.7.

Table 4. Comparison of $p$-frame energies for conjectured optimal configurations on $\mathbb{RP}^{d-1}$ and $\mathbb{CP}^{d-1}$ with LP lower bounds. Energies are evaluated at the odd integer midpoint of the conjectured optimality interval. $d$ | $\mathbb{F}$ | Energy | LP bound | $p$ | Name
---|---|---|---|---|---
$3$ | $\mathbb{R}$ | $0.1249$ | $0.1248$ | $7$ | icosahedron and dodecahedron
$4$ | $\mathbb{R}$ | $0.09628$ | $0.09607$ | $5$ |  $D_{4}$ root vectors
$5$ | $\mathbb{R}$ | $0.1183$ | $0.1170$ | $3$ | hemicube
$5$ | $\mathbb{R}$ | $0.06184$ | $0.06169$ | $5$ | Stroud design
$6$ | $\mathbb{R}$ | $0.09056$ | $0.08970$ | $3$ | cross-polytope and hemicube
$6$ | $\mathbb{R}$ | $0.04249$ | $0.04240$ | $5$ |  $E_{6}$ and $E_{6}^{*}$ roots
$7$ | $\mathbb{R}$ | $0.03065$ | $0.03060$ | $5$ |  $E_{7}$ and $E_{7}^{*}$ roots
$8$ | $\mathbb{R}$ | $0.05910$ | $0.05852$ | $3$ | mid-edges of regular simplex
$3$ | $\mathbb{C}$ | $0.01261$ | 0.01258 | $5$ | union equiangular lines
$5$ | $\mathbb{C}$ | $0.04200$ | 0.04184 | $5$ |  $O_{10}$ and $W(K_{5})$ roots

###  3.2. Properties of orthogonal polynomials

As already pointed out, for fixed $\Omega$, we write simply
$C_{n}(t)=C_{n}^{(\alpha,\beta)}(t)$. Recall that $C_{n}(1)=1$. In some of the
arguments in Section 3.4 we will use instead the monic polynomials
proportional to $C_{n}$; we therefore introduce notation
$Q_{n}(t)=Q_{n}^{(\alpha,\beta)}(t)$ for these Jacobi polynomials.

In this subsection we collect several results about orthogonal polynomials
relevant to the proof of our main theorem. Fix a space $\Omega$, and let
$\alpha$ and $\beta$ be the corresponding parameters of the associated Jacobi
polynomials. According to Proposition 2.3, a function being positive definite
on $\Omega$ is equivalent to having positive coefficients in the Jacobi
expansion in terms $Q_{n}^{(\alpha,\beta)}$.

It will be useful to consider adjacent Jacobi polynomials, defined as one of
the three sequences $Q^{k,l}_{n}=Q_{n}^{(\alpha+k,\beta+l)}$ with
$k,l\in\\{0,1\\}$, $k+l>0$. Specifically, we will need the following corollary
which comes out of representing $Q^{1,0}_{n}$ through $Q^{0,0}_{n}$ [Le1,
equation (3.4)]:

######  Proposition 3.2.

Adjacent Jacobi polynomials $Q_{n}^{1,0}$ are positive definite on $\Omega$.

On the other hand, adjacent polynomials $Q^{1,1}_{n}$, defined as orthogonal
with respect to the measure $(1-t^{2})\,d\nu^{(\alpha,\beta)}$, are not
positive definite. The following property, a special case of the strengthened
Krein condition [Le2, Lemma 3.22], can serve as a substitute.

######  Lemma 3.3.

$(t+1)Q^{1,1}_{n}(t)$ are positive definite on $\Omega$ for $n\geq 0$.

###### Proof.

For all $n\in\mathbb{N}_{0}$, $(t+1)Q^{1,1}_{n}$ is orthogonal to all
polynomials of degree less than $n$ with respect to the measure
$(1-t)d\nu^{(\alpha,\beta)}=c_{\alpha,\beta}d\nu^{(\alpha+1,\beta)}$, so it
can be expressed through the orthogonal polynomials corresponding to
$d\nu^{(\alpha+1,\beta)}$ as

$(t+1)Q^{1,1}_{n}(t)=Q^{1,0}_{n+1}(t)+bQ^{1,0}_{n}(t),$

for some constant $b$. Since all the roots of $Q^{1,0}_{n}$ lie in $(-1,1)$,
$\text{sgn\,}Q^{1,0}_{n}(-1)=(-1)^{n}$. Substituting $t=-1$ in the last
equation gives $Q^{1,0}_{n+1}(-1)+bQ^{1,0}_{n}(-1)=0$, and so $b\geq 0$. By
Proposition 3.2, each $Q^{1,0}_{n}(t)$ is positive definite, and thus
$(t+1)Q^{1,1}_{n}(t)$ is also positive definite. ∎

Lastly, we will need the strict positive-definiteness of polynomials
annihilated by subsets of roots of $p_{n}+\gamma p_{n-1}$. We recall the
following result.

######  Proposition 3.4 ([CK, Theorem 3.1]).

Given a sequence of orthogonal polynomials $p_{0}(t),$ $p_{1}(t),$
$p_{2}(t),\ldots$, let $t_{1}<\ldots<t_{n}$ be the zeros of $p_{n}+\gamma
p_{n-1}$ for some fixed $\gamma$. Then the polynomials

$\prod_{i=1}^{k}(t-t_{i}),\qquad 1\leq k<n,$

can be represented as a linear combination of
$p_{0}(t),p_{1}(t),\ldots,p_{n}(t)$ with positive coefficients.

###  3.3. Hermite interpolation

Let $f\in C^{K}[a,b]$, for some $K\in\mathbb{N}_{0}$, and let a collection
$t_{1}<\ldots<t_{m}\subset[a,b]$, as well as positive integers
$k_{1},\ldots,k_{m}$ be given with

$\max\\{k_{1},\ldots,k_{m}\\}\leq K+1.$

There exists a polynomial $p$ of degree less than $D=\sum_{i=1}^{m}k_{i}$,
such that for $1\leq i\leq m$ and $0\leq k<k_{i}$,

$p^{(k)}(t_{i})=f^{(k)}(t_{i}).$

Such a $p$ is called the Hermite interpolating polynomial of $f$; it always
exists and is unique because the linear map that takes a polynomial $p$ of
degree less than $D$ to

$(p(t_{1}),p^{\prime}(t_{1}),\ldots,p^{(k_{1}-1)}(t_{1}),p(t_{2}),p^{\prime}(t_{2}),\ldots,p^{k_{m}-1}(t_{m}))$

is bijective.

It is convenient to organize both the collection $t_{1}<\ldots<t_{m}$ and the
orders of derivatives $k_{1},\ldots,k_{m}$ into a polynomial $g(t)$. Given
such a polynomial

$g(t)=\prod_{i=1}^{m}(t-t_{i})^{k_{i}},$

where $D=\deg(g)\geq 1$, we write $H\left[f,g\right]$ for the interpolating
polynomial of degree less than $D$ that agrees with $f$ at each $t_{i}$ to the
order $k_{i}$. Similarly, we let

$Q[f,g](t)=\frac{f(t)-H\left[f,g\right](t)}{g(t)},$

be the divided difference associated with the polynomial $g$. Under the above
hypotheses, for every $t\in[a,b]$ and a collection $t_{1}<t_{2}<\ldots<t_{m}$
as above, there exists $\xi\in(a,b)$ such that
$\min(t,t_{1})<\xi<\max(t,t_{m})$, and

(3.1) $Q[f,g](t)=\frac{f^{(D)}(\xi)}{D!}.$

Enumerate the roots of $g$ with multiplicities in increasing order, and denote
these by $s_{j},1\leq j\leq D$, where $s_{j}\leq s_{j+1}$. Let $g_{n}$ be the
polynomial annihilated on the first $n$ elements of the sequence
$s_{1},\ldots,s_{D}$:

$g_{n}(t)=\prod_{j=1}^{n}(t-s_{j}),\qquad 1\leq n\leq D.$

The usual assignment of the empty product applies here: $g_{0}(t)=1$.

By the Newton’s formula [DL, Chapter 4.6–7], the Hermite interpolating
polynomial $H\left[f,g\right]$ can be represented as

(3.2)
$H\left[f,g\right](t)=f(s_{1})+\sum_{j=1}^{D-1}g_{j}(t)\,Q[f,g_{j}](s_{j+1}).$

The relevant property of the $p$-frame kernel
$\left(\frac{s+1}{2}\right)^{p/2}$ considered on a projective space
$\mathbb{FP}^{d-1}$, is that its first several derivatives are nonnegative on
$(-1,1)$, followed by a negative one. Positivity of the derivatives implies,
due to (3.1), that the divided differences in the formula (3.2) for the
$p$-frame kernel are nonnegative. It will be convenient to introduce notation
for this number of nonnegative derivatives of a function.

######  Definition 3.5.

Let $f\in C^{M}(a,b)$. We say that $f$ is absolutely monotonic of degree $M$
if $f^{(k)}(t)\geq 0$ for $0\leq k\leq M$ and $t\in(a,b)$.

Compare the above definition with that of absolutely monotonic functions,
where all derivatives of a function are non-negative. Usefulness of this
pattern of signs of the derivatives lies in that the Hermite interpolant of an
absolutely monotonic function $f$ of degree $M$ with $(M+1)$st derivative
negative, will stay below $f$, as shown in the following observation [Y1].

######  Lemma 3.6.

Let $f:[-1,1]\to\mathbb{R}$ be absolutely monotonic of degree $M$, and
$f^{(M+1)}(t)\leq 0$ for all $t\in(-1,1)$. If the roots of a polynomial $g$ of
degree $M+1$ are contained in $[-1,1]$, and in addition $g(t)\leq 0$ for
$t\in[-1,1]$, then,

$f(t)\geq H[f,g](t),\qquad t\in[-1,1].$

###### Proof.

According to (3.1), there exists $\xi\in(-1,1)$ such that
$\min(t,t_{0})<\xi<\max(t,t_{M})$, where the roots of $g$ are
$t_{0}\leq\ldots\leq t_{M}$, and

$f(t)-H[f,g](t)=\frac{f^{(M+1)}(\xi)}{(M+1)!}g(t).$

The expression on the right is nonnegative, so the conclusion of the lemma
follows. ∎

###  3.4. Optimality of tight designs

As above, $\Omega$ is a compact, connected two-point homogeneous space and
$Q_{0},Q_{1},Q_{2},\ldots$ are the corresponding orthogonal polynomials.
Recall that $Q_{n}$ are orthogonal with respect to the measure
$d\nu^{(\alpha,\beta)}=\frac{1}{\gamma_{\alpha,\beta}}(1-t)^{\alpha}(1+t)^{\beta}dt$,
where the parameters $\alpha$, $\beta$ are chosen as in Section 2.1. The main
result of this section is the following.

######  Theorem 3.7.

Let $f$ be absolutely monotonic of degree $M$, with $f^{(M+1)}(t)\leq 0$ for
$t\in(-1,1)$. Then for a tight $M$-design $\mathcal{C}$,

$\mu_{\mathcal{C}}=\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\delta_{x}$

is a minimizer of

$I_{f}(\mu)=\int_{\Omega}\int_{\Omega}f(\tau(x,y))\,d\mu(x)d\mu(y)$

over $\mathcal{P}(\Omega)$, the set of probability measures on $\Omega$.

In what follows we give a proof of the above theorem, splitting it into two
separate cases, depending on whether the code $\mathcal{C}$ contains two
points separated by the diameter of $\Omega$; equivalently, depending on the
parity of the degree of $\mathcal{C}$, $M$.

######  Proposition 3.8.

Theorem 3.7 holds when $M=2m$, $m\geq 1$.

###### Proof.

Let $t_{1}<\ldots<t_{m}<t_{m+1}=1$ be the values of
$\tau(x,y)=\cos(\vartheta(x,y))$ occurring in $\mathcal{C}$. Let further

$g_{k}(t)=\prod_{i=1}^{k}(t-t_{i}),\qquad 1\leq k\leq m+1.$

and

$g(t)=g_{m}(t)\,g_{m+1}(t)=(t-1)g^{2}_{m}(t).$

To prove the statement of the theorem, we verify the following chain of
inequalities, satisfied for arbitrary $\mu\in\mathcal{P}(\Omega)$, similar to
the proof of Lemma 3.1,

(3.3) $I_{f}(\mu)\geq I_{H[f,g]}(\mu)\geq
I_{H[f,g]}(\sigma)=I_{H[f,g]}(\mu_{\mathcal{C}})=I_{f}(\mu_{\mathcal{C}}).$

The equality $I_{H[f,g]}(\sigma)=I_{H[f,g]}(\mu_{\mathcal{C}})$ follows since
$\mathcal{C}$ is a design of degree $2m\geq\deg H[f,g]$. The last equality
holds since $H[f,g]$ agrees with $f$ at the cosines of distances occurring in
$\mathcal{C}$. Since $g(t)\leq 0$ for $t\in[-1,1]$, Lemma 3.3 implies that
$f(t)\geq H[f,g](t)$, $t\in[-1,1]$, which gives the first inequality. It
remains to show the second inequality: it will follow from the positive
definiteness of $H[f,g]$, which we will now demonstrate.

For any $n<m$, the degree of $g_{m+1}(t)Q_{n}(t)$ is at most $2m$. As
$\mathcal{C}$ is a $2m$-design, for every fixed $y\in\mathcal{C}$ there holds

$\displaystyle\int_{-1}^{1}g_{m+1}(t)Q_{n}(t)d\nu^{(\alpha,\beta)}$
$\displaystyle=\int_{\Omega}g_{m+1}(\tau(x,y))Q_{n}(\tau(x,y))d\sigma(x)$
$\displaystyle=\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}g_{m+1}(\tau(x,y))Q_{n}(\tau(x,y))$
$\displaystyle=\frac{1}{|\mathcal{C}|}\sum_{i=1}^{m+1}c_{i}g_{m+1}(t_{i})Q_{n}(t_{i})$
$\displaystyle=0,$

since, by construction, $g_{m+1}$ is annihilated on all the $t_{i}$. The
constants $c_{i}$ are given by

$c_{i}=|\\{x\in\mathcal{C}\ |\ \tau(x,y)=t_{i}\\}|.$

Both $g_{m+1}$ and $Q_{m+1}$ are monic, so we conclude that

$g_{m+1}(t)=Q_{m+1}(t)+\gamma Q_{m}(t),$

for some $\gamma\in\mathbb{R}$. By Proposition 3.4, subproducts of zeros of
$g_{m+1}$, which we denote by $g_{k}$, $1\leq k\leq m$, can be expressed as
linear combinations of $Q_{n}$ with positive coefficients, and therefore are
positive definite.

According to the Newton’s formula (3.2), the Hermite interpolant of $f$ can be
expressed as the sum of partial products of factors of $g$ multiplied by the
appropriate divided difference. We will use this formula to show that $H[f,g]$
is positive definite. Indeed, (3.2) gives

$H\left[f,g\right](t)=f(t_{1})+\sum_{k=1}^{m}\bigg{(}g_{k}(t)g_{k-1}(t)\,Q\left[f,g_{k}g_{k-1}\right]\left(t_{k}\right)+g^{2}_{k}(t)\,Q\left[f,g_{k}^{2}\right]\left(t_{k+1}\right)\bigg{)},$

where as usual, $g_{0}=1$. Observe that the divided differences in the last
equation are nonnegative due to (3.1), as the function $f$ is absolutely
monotonic of degree $2m$. Since we have shown that each $g_{k}$ is positive
definite, Schur’s theorem implies that so are $g_{k}^{2}$ and $g_{k}g_{k+1}$,
and it follows that $H[f,g]$ is positive definite as well.

∎

Before turning to the proof of Theorem 3.7 for tight designs of odd strength,
recall the definition of the adjacent polynomials
$Q^{1,1}_{n}=Q_{n}^{(\alpha+1,\beta+1)}$ for $n\geq 0$. They are monic,
orthogonal with respect to the measure

$d\nu^{({\alpha+1},{\beta+1})}(t)=\frac{1}{\gamma_{\alpha+1,\beta+1}}(1-t)^{\alpha+1}(1+t)^{\beta+1}dt=\frac{\gamma_{\alpha,\beta}}{\gamma_{\alpha+1,\beta+1}}(1-t^{2})d\nu^{({\alpha},{\beta})}(t),$

since the polynomials $Q_{n}^{(\alpha,\beta)}(t)$ are orthogonal with respect
to measure $d\nu^{({\alpha},{\beta})}$.

######  Proposition 3.9.

Theorem 3.7 holds when $M=2m-1$, $m\geq 1$.

###### Proof.

Suppose that $\mathcal{C}\subset\Omega$ is a tight $(2m-1)$-design. As
discussed in Section 2.3 tight designs of odd strength necessarily contain
antipodal points, i.e. there exist $x$, $y\in\mathcal{C}$ such that
$\vartheta(x,y)=\pi$ and thus
$-1\in\mathcal{A}(\mathcal{C})=\\{\tau(x,y)|\,x,y\in\mathcal{C}\\}$. Let
$-1=t_{1}<\ldots<t_{m}<t_{m+1}=1$ be the values of $\tau(\vartheta(x,y))$ for
$x,y\in\mathcal{C}$. Let further

$w(t)=\prod_{j=2}^{m}(t-t_{j})$

and

$g(t)=w^{2}(t)(t^{2}-1).$

As in the proof of Proposition 3.8, we need to verify the inequalities (3.3).
Applying Lemma 3.3 to $H[f,g]$ gives the first inequality; it remains to show
positive-definiteness of $H[f,g]$.

For $n<m-1$, the degree of $(1-t^{2})w(t)Q^{1,1}_{n}(t)$ is at most $2m-1$, so
for any $y\in\mathcal{C}$ there holds

$\displaystyle\frac{\gamma_{\alpha+1,\beta+1}}{\gamma_{\alpha,\beta}}\int_{-1}^{1}w(t)Q^{1,1}_{n}(t)d\nu^{(\alpha+1,\beta+1)}$
$\displaystyle=\int_{\Omega}(1-\tau^{2}(x,y))w(\tau(x,y))Q^{1,1}_{n}(\tau(x,y))d\sigma(x)$
$\displaystyle=\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}(1-\tau^{2}(x,y))w(\tau(x,y))Q^{1,1}_{n}(\tau(x,y))$
$\displaystyle=\frac{1}{|\mathcal{C}|}\sum_{j=1}^{m+1}c_{j}(1-t_{j}^{2})w(t_{j})Q^{1,1}_{n}(t_{j})$
$\displaystyle=0,$

as $(1-t^{2})w(t)$ is annihilated on the cosines of distances from
$\mathcal{C}$. Because $w(t)$ is a degree $m-1$ monic polynomial, the above
implies $w(t)=Q^{1,1}_{m-1}(t)$. By Proposition 3.4, this also means that for
$2\leq k\leq m-1$, polynomials $\prod_{j=2}^{k}(t-t_{j})$ are linear
combinations of $Q^{1,1}_{n}$ with nonnegative coefficients. Since the cone of
functions with nonnegative Jacobi coefficients with respect to $Q^{1,1}_{n}$
is closed under multiplication, polynomials $\prod_{j=2}^{k}(t-t_{j})^{2}$ and
$(t-t_{k})\prod_{j=2}^{k-1}(t-t_{j})^{2}$ also have nonnegative Jacobi
coefficients in $Q^{1,1}_{n}$. Due to Lemma 3.3, since $t-t_{1}=t+1$, we
obtain that

(3.4)
$a_{k}(t):=(t-t_{1})(t-t_{l})\prod_{j=2}^{k-1}(t-t_{j})^{2}\quad\text{and}\quad
b_{k}(t):=(t-t_{1})\prod_{j=2}^{k}(t-t_{j})^{2},$

are linear combinations of $Q_{n}^{(\alpha,\beta)}$ with positive
coefficients, that is, they are positive definite on $\Omega$ for $1\leq k\leq
m$.

We conclude by the same observations as in the proof of Proposition 3.8; in
particular, the positive definiteness of the Hermite interpolant $H[f,g]$
follows from the representation

$H\left[f,g\right](t)=f(t_{1})+\sum_{k=2}^{m-1}\bigg{(}a_{k}(t)\,Q\left[f,a_{k}\right]\left(t_{k}\right)+b_{k}(t)\,Q\left[f,b_{k}\right]\left(t_{k+1}\right)\bigg{)},$

combined with the absolute monotonicity of $f$ to degree $2m-1$, which implies
positivity of the divided differences $Q$. ∎

######  Example 3.10.

As an example of another application of Theorem 3.7, consider the case that
$f(t)=a+bt+ct^{2}+dt^{3}$ is given as potential function. In this case, some
elementary considerations show that if

  1. (i)

$d\leq 0$,

  2. (ii)

$c\geq-3d$,

  3. (iii)

$c^{2}-3bd\geq 0$,

  4. (iv)

$-c-\sqrt{c^{2}-3bd}\leq 3d$, and,

  5. (v)

$-c+\sqrt{c^{2}-3bd}\geq 3d$,

then $f$ is absolutely montonic of degree $2$ up to a constant. Hence, any
potential function of the above form has as minimizer of the $f$-energy on any
of the projective spaces a tight $2$-design. In particular, for $f$ as above,
the icosahedron is a minimizer of energy integral $I_{f}(\mu)$ over symmetric
measures on the sphere $\mathbb{S}^{2}$. Note that the constant term can be
ignored, so it suffices to only consider the sign of derivatives. In
particular, if $b>0$ and $d$ becomes sufficiently small in magnitude, the
above inequalities will hold.

For comparison, on $\mathbb{S}^{2}$, $f(\tau(x,y))=f(2|\langle
x,y\rangle|^{2}-1)$ is positive definite (up to a constant), precisely when
$\widehat{f}_{1},\widehat{f}_{2},$ and $\widehat{f}_{3}$ are positive, or
equivalently (by calculation),

  1. (i)

$4b-2c+3d\geq 0$,

  2. (ii)

$2c-d\geq 0$, and,

  3. (iii)

$d\geq 0$.

Thus, $I_{f}$ is minimized by the surface measure $\sigma$ precisely for the
coefficients satisfying the above inequalities.

###  3.5. Uniqueness of minimizers supported on tight designs

The proofs in the last section left the question of uniqueness of minimizers
open. Are there any other minimizers for $p$-frame energies when tight designs
minimize and $p$ is not an even integer? The answer, as this section details,
is no.

Whenever a tight design minimizes $I_{f}$, any minimizer for an energy with
kernel $f$ that is absolutely monotonic to degree $M$, and which satisfies
$f^{(M+1)}(t)<0$, $t\in(-1,1)$, is minimized only on a tight design, although
such designs are not necessarily unique up to equivalence (see Section 9 for
more details).

To prove uniqueness, up to tightness, we consider the spherical and projective
cases separately, although within the same general framework. Let
$\mathcal{C}$ be a finite $m$-distance set in $\Omega=\mathbb{S}^{d-1}$ or
$\mathbb{FP}^{d-1}$ with $\mathcal{A}(\mathcal{C})=\\{\tau(x,y)\ |\
x,y\in\mathcal{C}\\}$, so that $m=|\mathcal{A}(\mathcal{C})|-1$. Set
$e=|\mathcal{A}(\mathcal{C})\setminus\\{-1\\}|-1$ and $\epsilon$ to be $1$ if
$-1\in\mathcal{A}(\mathcal{C})$ and $0$ otherwise, i.e. $e=m-\epsilon$. The
annihilating polynomial of a configuration $Ann(\mathcal{C})$ is defined by
$\Pi_{\alpha\in\mathcal{A}(\mathcal{C})}(x-\alpha)$. For a positive number
$t$, let $(t)_{k}=t(t+1)\dots(t+k-1)$ be the Pochhammer symbol, and let
$\mathcal{T}$ be a tight design of size $N$ in $\Omega$. The following lemmas
from [DGS] and [H1] provide some additional properties of tight designs which
will be useful in the theorem which follows.

######  Lemma 3.11.

Let $\mathcal{C}$ be an $m$-distance configuration,
$\mathcal{C}\subset\mathbb{S}^{d-1}$, $|\mathcal{C}|=N$

  1. (i)

If $\mathcal{C}$ is a $t$-design, then $t\leq 2m$ and
$N\leq\binom{d+m-1}{d-1}+\binom{d+m-2}{d-1}$. Equality holds in either of
these inequalities if and only if $\mathcal{C}$ is a tight $2m$-design and
$Ann(\mathcal{C})=Ann(\mathcal{T})$.

  2. (ii)

If $\mathcal{C}$ is an antipodal $t$-design, then $t\leq 2m-1$, and
$N\leq\binom{d+m-2}{d-1}$. Equality holds in either of these inequalities if
and only if $\mathcal{C}$ is a tight $(2m-1)$-design, and in particular
$Ann(\mathcal{C})=Ann(\mathcal{T})$.

The projective analog of the above lemma is now given.

######  Lemma 3.12.

For $m$-distance configuration $\mathcal{C}\subset\mathbb{FP}^{d-1}$,
$|\mathcal{C}|=N$,

  1. (i)

$N\leq\frac{(\alpha+1)_{m}(\alpha+1-\beta)_{\epsilon}}{(\beta+1)_{m}e!}$

and equality holds if and only if $\mathcal{C}$ is a tight $(2m-1)$-design and
$Ann(\mathcal{C})=Ann(\mathcal{T})$.

  2. (ii)

If $\mathcal{C}$ is a $t$-design but not a $(t+1)$-design, then $t\leq m+e$
with equality if and only if equality holds in part one of this lemma.

The annihilating polynomials for tight designs can be worked out explicitly
and are given in [H1]. We now show uniqueness of minimizing measures in
Theorem 3.7.

######  Theorem 3.13.

Suppose that a tight $M$-design, $\mathcal{C}$, minimizes the $f$-energy
integral, for $f$ absolutely monotonic of degree $M$ and such that
$f^{(M+1)}(t)<0$, $t\in(-1,1)$. Then any minimizer of $I_{f}(\mu)$ must be a
tight $M$-design.

###### Proof.

The argument developed to prove Theorem 3.7 may be described concisely through
the following string of inequalities

$I_{f}(\mu)\geq I_{h}(\mu)\geq I_{h}(\sigma)=I_{h}(\nu)=I_{f}(\nu).$

In order for $I_{f}(\mu)=I_{f}(\nu)$ to hold, the first inequalities must be
equalities. The first inequality can only be sharp in the case that the values
$\tau(x,y)$, $x,y\in\operatorname{supp}{\mu}$ are a subset of those given for
$x,y\in\operatorname{supp}{\xi}$, where $\xi$ is the minimizing tight design.
This follows from the fact that $h(t)<f(t)$ for all $t$ outside this set by
the remainder formula from Lemma 3.6. In particular, this gives that
$|\operatorname{supp}\mu|$ is finite.

The second inequality is sharp only when $\mu$ is a weighted design of at
least the order corresponding to the minimizing tight design. By the above
lemma $\mu$ then satisfies
$\mathcal{A}(\operatorname{supp}\mu)\subset\mathcal{A}(\operatorname{supp}\xi)$.
Since tight designs maximize the size of a code over all $m$-distance sets,
where $m=|\mathcal{A}(\operatorname{supp}\xi)|$, finally it holds that
$|\mathcal{A}(\operatorname{supp}\mu)|=|\mathcal{A}(\operatorname{supp}\xi)|$
and $|\operatorname{supp}\mu|=|\operatorname{supp}\xi|$, so that $\mu$ must be
tight. ∎

##  4\. Optimality of the 600-cell

This section concerns only the $p$-frame kernels; it will be shown here that
the 600-cell minimizes the $p$-frame energy on $\mathbb{S}^{3}$ for a certain
range of $p$. The $600$-cell is one of the six $4$-dimensional convex regular
polytopes; it has $600$ tetrahedral faces, which explains the origin of its
name. When its 120 vertices are identified with unit quaternions, they give a
representation of the elements of a group known as the binary icosahedral
group [St].

As discussed above (2.4), optimization of $p$-frame energy on the sphere
$\mathbb{S}^{3}$ is equivalent to optimization of the expression
$\iint_{\mathbb{RP}^{3}}f(\tau(x,y))\,d\mu(x)d\mu(u)$ over measures $\mu$ on
$\mathbb{RP}^{3}$, where the kernel $f$ is given by

$f(t)=\bigg{(}\frac{1+t}{2}\bigg{)}^{\frac{p}{2}}.$

We therefore assume for the rest of this section the underlying space to be
$\mathbb{RP}^{3}$, and use the corresponding Jacobi polynomials
$C_{n}^{(-1/2,1/2)}(t)$. Following the approach of the previous section, we
will establish a sequence of inequalities similar to (3.3).

The $600$-cell is only a projective $5$-design and therefore not tight, cf.
Table 7. The authors in [CK], motivated by an approach found in the paper [A],
found means to prove universal optimality of the $600$-cell by using a higher
degree interpolating polynomial. The $600$-cell has the notable property that
$7$th, $8$th, and $9$th degree harmonic averages over it vanish, although the
$6$th degree average does not. This allows for constructing a degree $8$
polynomial $h$ which is less than or equal to $f$, positive definite, and
agrees with $f$ at the distances appearing in the $600$-cell, and which
finally has the property that its $6$th Jacobi coefficient vanishes.

For a polynomial $h$ of the form,

(4.1) $h=\sum_{\begin{subarray}{c}n=0\\\ n\neq
6\end{subarray}}^{8}\widehat{h}_{n}\,C_{n}^{\left(1/2,-1/2\right)}(t),$

the coefficients $\widehat{h}_{n}$ can be uniquely determined as functions of
$p$ by setting

$\displaystyle h(t_{i})=$ $\displaystyle f(t_{i}),\qquad 1\leq i\leq 5$
$\displaystyle h^{\prime}(t_{i})=$ $\displaystyle f^{\prime}(t_{i}),\qquad
2\leq i\leq 4,$

where $-1=t_{1}<t_{2}<\ldots<t_{5}=1$ are the values of $\tau(x,y)$ when
vectors $x,y$ vary over the vertices of the 600-cell, see the proof of Theorem
4.2 below and Table 7. It turns out that for all $p\in[8,10]$,
$\widehat{h}_{n}(p)\geq 0$ when $0\leq n\leq 8,n\neq 6$. We apply a computer-
assisted approach to verify this positivity; specifically, using interval
arithmetic, we compute values of $\widehat{h}_{n}(p)$ on a grid fine enough to
guarantee that $\widehat{h}_{n}(p)\geq 0$. The details of this computation are
available as an html file and as a SageMath [S] notebook, distributed with the
arXiv submission of this paper.

Even though the computations performed by our program are carried out in
finite floating point precision, interval arithmetic guarantees that the
results of these computations lie in precisely defined intervals. SageMath
provides an implementation of the interval arithmetic through a wrapper of the
MPFI [MPFI] library [ZC+, S]. The computer-assisted argument yields the
following.

######  Lemma 4.1.

If $p\in[8,10]$ and the polynomial $h$ is constructed as above, the
coefficients $\widehat{h}_{n}$ in the Jacobi expansion (4.1) satisfy
$\widehat{h}_{n}(p)\geq 0$.

Using this fact we show optimality of the $600$-cell on the range
$p\in[8,10]$.

######  Theorem 4.2.

The $600$-cell minimizes the $p$-frame energy for $p\in[8,10]$ over Borel
probability measures on $\mathbb{S}^{3}$ or $\mathbb{RP}^{3}$.

###### Proof.

Let $f(t)=\left(\frac{t+1}{2}\right)^{p/2}$ for some $8<p<10$, $t_{1}=-1$,
$t_{2}=\frac{-\sqrt{5}-1}{4}$, $t_{3}=-\frac{1}{2}$,
$t_{4}=\frac{\sqrt{5}-1}{4}$, and $t_{5}=1$. Let $h(t)$ be the $8$th degree
polynomial given by (4.1), such that $h(t_{i})=p(t_{i})$ for $1\leq i\leq 5$,
and $h^{\prime}(t_{i})=p^{\prime}(t_{i})$ for $2\leq i\leq 4$. By Lemma 4.1,
the coefficients $\widehat{h}_{n}$ are non-negative for $p\in[8,10]$.

Let $p(t)=(t^{2}-1)\prod_{i=2}^{4}(t-t_{i})^{2}$ and $\tilde{h}(t)=H[f,p](t)$.
Then we also have $\tilde{h}(t)=H[h,p](t)$. This gives

$f(t)-\tilde{h}(t)=\frac{f^{(8)}(\xi)}{8!}p(t)\geq 0,$

and

$h(t)-\tilde{h}(t)=\frac{h^{(8)}(\nu)}{8!}p(t)\leq 0.$

We thus have $f(t)-h(t)=f(t)-\tilde{h}(t)+\tilde{h}(t)-h(t)\geq 0$. Since
$h(t)$ is positive definite and $\widehat{h}_{6}=0$, for the 600-cell
$\mathcal{C}_{600}$, we have the following sequence of inequalities

$I_{f}(\mu)\geq I_{h}(\mu)\geq
I_{h}(\sigma)=I_{h}(\mu_{\mathcal{C}_{600}})=I_{f}(\mu_{\mathcal{C}_{600}}),$

implying that equally weighted vertices of $\mathcal{C}_{600}$ minimize
$p$-frame energy. ∎

##  5\. Conjectured minimizers

###  5.1. New small weighted projective design

We now collect facts on the $85$ vector system which was found while
numerically minimizing the $p=5$ frame potential in $\mathbb{C}^{5}$. This
system of vectors forms a weighted design of strength $3$, or equivalently,
for the functional $\sum_{i,j}|\langle
v_{i},v_{j}\rangle|^{6}\omega_{i}\omega_{j}$, the weighted system takes the
value $1/35$, thus minimizing this quantity over all probability measures
$\mu=\sum_{i}\delta_{v_{i}}\omega_{i}$, $\sum_{i}\omega_{i}=1$ supported on
unit vectors $\|v_{i}\|=1$ in $\mathbb{C}^{5}$ [We]. The above construction
appears to be new especially when comparing its size to previously obtained
bounds from [LS] for smallest known $3$ weighted designs in $\mathbb{C}^{5}$.

One part of the system is well studied, given by the root vectors
corresponding to the $45$ 2-reflections which generate the unitary reflection
group $W(K_{5})$ of $51840$ elements [LT]. This group is alternatively
described as the group $G_{3}(10)\simeq(C_{6}\times SU_{4}(2)):C_{2}$, one of
the maximal finite irreducible subgroups of $GL_{10}(\mathbb{Z})$ [So].
$SU_{4}(2)$ here is just the special linear group of $4\times 4$ matrices,
unitary matrices over $\mathbb{F}_{2^{2}}$, with determinant one.

Choosing the representation of the root vectors in $W(K_{5})$ as
$X_{1}=\\{\sigma((1,0,0,0,0))\\}\cup\\{\sigma(\frac{1}{2}(0,1,\pm\omega,\pm\omega,\pm
1))\\}$ under cyclic coordinate permutations, $\sigma$, the new weighted
design arises when this system is joined with some other 40 vectors. The
second system may be described as
$\Psi=\\{\sigma(\frac{1}{\sqrt{3}}(1,0,\pm\omega,\pm\omega,0))\\}\cup\\{\sigma(\frac{1}{\sqrt{3}}(1,\pm\omega,\pm
1,0,0))\\}$ also generated under cyclic coordinate permutations. The
projective design is finally given by assigning weights to the $W(K_{5})$
system joined with the $40$ vector system after giving $\Psi$ the orientation
$X_{2}=U\Psi$, where

(5.1) $U=\frac{1}{2}\left[\begin{array}[]{ccccc}1&-\omega&-\omega&1&0\\\
-1&1&-\omega^{2}&0&-\omega^{2}\\\ \omega^{2}&0&-\omega^{2}&1&1\\\
0&1&\omega&-\omega&-1\\\
\omega^{2}&\omega&0&-\omega&\omega^{2}\end{array}\right],$

is unitary ($\omega=e^{2\pi i/3}$). With the above orientation the $40$ points
in $X_{2}$ appear to fit so that each point is a maximizer of the projective
distance from each of the $45$ vectors in the $W(K_{5})$ system and vice
versa. If so, the additional $40$ points satisfy that they are the points at
greatest distance from the original $45$, in particular.

To form a weighted $3$-design, the corresponding weights for $X_{1}$, the $45$
vector system, are $\omega_{1}=\frac{4}{315}$, and for the remaining $40$
vectors in $X_{2}$, the weights are $\omega_{2}=\frac{3}{280}$. In total the
distribution of absolute values of inner products that appears in the
unweighted $85$ vector system is given in Table 5. The supplementary files in
the arXiv version of this manuscript provide a magma script which verifies
that $\sum_{i,j}|\langle v_{i},v_{j}\rangle|^{6}\omega_{i}\omega_{j}=1/35$, so
that the system is a projective $3$-design. This script can additionally be
used to show the automorphism group of the above system of $85$ vectors is
isomorphic to the group $SU_{4}(2)$ of order
$|SU_{4}(2)|=25920=2^{6}\cdot(2^{4}-1)\cdot(2^{3}+1)\cdot(2^{2}-1)$, through
use of a library from [HW].

Table 5. Table of inner products between vectors in parts $X_{1},X_{2}$ of the new cubature formula of $85$-vectors. $N$ counts the number of times a value occurs as an entry in $|X_{i}^{\prime}X_{j}|$, $i,j=1,2$. | $|\langle x,y\rangle|$ | $N$
---|---|---
$|X_{1}^{\prime}X_{1}|$ | $0,1/2,1$ | $540,1440,45$
$|X_{2}^{\prime}X_{2}|$ | $1/3,1/\sqrt{3},1$ | $1080,480,40$
$|X_{1}^{\prime}X_{2}|$ | $0,1/\sqrt{3}$ | $720,1080$
$|X_{2}^{\prime}X_{1}|$ | $0,1/\sqrt{3}$ | $720,1080$

The above construction hides the relation between its two parts. The $85$
vectors in $\mathbb{C}^{5}$ may be seen, after canonically embedding the
vectors in $\mathbb{R}^{10}$, as the weighted union of vectors coming from two
$10$ dimensional lattices. Under this identification, the $45$ vectors in the
$W(K_{5})$ system may be selected as, up to projective equivalence (modulo
multiples of sixth roots of unity), the $270$ minimal vectors of the lattice
called $(C_{6}\times SU_{4}(2)):C_{2}$ in the database [NS], and the other
$40$ points are taken one from each antipodal pair of the $80$ minimal vectors
of the shorter Coxeter-Todd lattice, $O_{10}$ detailed in [RS]. The
relationship between these two lattices is that $(C_{6}\times
SU_{4}(2)):C_{2}$ is similar to the maximal even sub-lattice of $O_{10}$. In
our tables, we choose to name these the $W(K_{5})$ and $O_{10}$ lattices. We
prefer an alternative name for the first since the automorphism group of each
lattice is $(C_{6}\times SU_{4}(2)):C_{2}$.

Altogether, upon splitting the weights across minimal vectors in appropriately
scaled and oriented copies of these lattices and then complexifying
everything, one arrives at the cubature formula, which when viewed
projectively, is a system of $85$ vectors improving on the best previous known
bound of size $320$ for such a formula (see [Sh]). Some experiments suggest
this might be the smallest sized weighted projective $3$-design in
$\mathbb{CP}^{4}$. Expecting that this code might be optimal in a few other
settings, we conjecture:

######  Conjecture 5.1.

The code constructed in this section of $85$ points in $\mathbb{C}^{5}$ is
universally optimal.

###  5.2. Other weighted designs

####  5.2.1. $11$ points in $\mathbb{R}^{3}$

It seems that as $p$ goes to $6$ from below, the limiting minimizing
configuration on the sphere $\mathbb{S}^{2}$ is of the following form.
Concisely, the system consists of all combinations of signs of the $6$ vectors
below,

$\left[\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\ 0&0&1\\\
\frac{2}{\sqrt{7}}&\sqrt{\frac{3}{7}}&0\\\
\frac{2}{\sqrt{7}}&0&\sqrt{\frac{3}{7}}\\\
\sqrt{\frac{1}{7}}&\sqrt{\frac{3}{7}}&\sqrt{\frac{3}{7}}\end{array}\right]$

with the weights,

$\frac{2}{27},\frac{1}{10},\frac{1}{10},\frac{49}{540},\frac{49}{540},\frac{49}{540}$

on each line. The off-diagonal inner products are then

$1/7,-1/7,5/7,-5/7,\sqrt{3/7},-\sqrt{3/7},0,\sqrt{1/7},-\sqrt{1/7},4/7,-4/7,\sqrt{4/7},-\sqrt{4/7}$

appearing in number, $(10,18,10,10,14,10,14,6,2,4,4,6,2)$ respectively. From
these facts, one may check that the $11$ lines defined by these vectors forms
a projective $3$-design. Notably, this is the same extremal code, which forms
a minimal cubature formula and is found also in [Rez, page 135].

####  5.2.2. $16$ points in $\mathbb{R}^{3}$

Lines through antipodal points in the union of a regular icosahedron with its
dual dodecahedron. The frequencies of absolute values of inner products are
$N(\sqrt{\frac{1}{15}(5-2\sqrt{5})})=60,$
$N(\frac{\sqrt{75+30\sqrt{5}}}{15})=60,$ $N(\frac{1}{3})=60,$
$N(\frac{1}{\sqrt{5}})=30,$ $N(\sqrt{\frac{5}{9}})=30,$ and $N(1)=60$. The
weights making this configuration a projective $4$-design are
$\omega_{1}=5/84$ and $\omega_{2}=9/140$ for the icosahedron and dodecahedron
vertices respectively.

####  5.2.3. $11$ points in $\mathbb{R}^{4}$

See Table 6 for what appears to be the limiting minimizing configuration as
$p$ goes to $6$ from below when minimizing over $\mathbb{S}^{3}$.

Table 6. The Gram matrix of the weighted projective $2$-design in
$\mathbb{R}\mathbb{P}^{3}$ which appears as a minimizer as $p\rightarrow
4^{-}$ along with ordered weights, with each weight corresponding to the
vector with inner products given in the adjacent row. In the matrix, $a$ and
$b$ are $\frac{\sqrt{5}+1}{6}$ and $\frac{1}{6}\sqrt{(6-2\sqrt{5})}$,
respectively.

$\displaystyle\left[\begin{array}[]{ccccccccccc|c}\rule{0.0pt}{12.91663pt}\
1&-\frac{2}{3}&a&a&a&a&b&b&b&b&\frac{\sqrt{6}}{6}&\frac{3}{40}\\\
\rule{0.0pt}{12.91663pt}-\frac{2}{3}&\
1&-b&-b&-b&-b&-a&-a&-a&-a&\frac{\sqrt{6}}{6}&\frac{3}{40}\\\
\rule{0.0pt}{12.91663pt}\ a&-b&1&\ \frac{1}{3}&-\frac{1}{3}&\
\frac{1}{3}&-\frac{\sqrt{2}}{3}&-\frac{\sqrt{2}}{3}&\frac{\sqrt{2}}{3}&\frac{\sqrt{2}}{3}&\
\frac{\sqrt{6}}{6}&\frac{3}{32}\\\ \rule{0.0pt}{12.91663pt}\ a&-b&\
\frac{1}{3}&\ 1&\
\frac{1}{3}&-\frac{1}{3}&\frac{\sqrt{2}}{3}&-\frac{\sqrt{2}}{3}&\frac{\sqrt{2}}{3}&-\frac{\sqrt{2}}{3}&\
\frac{\sqrt{6}}{6}&\frac{3}{32}\\\ \rule{0.0pt}{12.91663pt}\
a&-b&-\frac{1}{3}&\ \frac{1}{3}&\ 1&\
\frac{1}{3}&\frac{\sqrt{2}}{3}&\frac{\sqrt{2}}{3}&-\frac{\sqrt{2}}{3}&-\frac{\sqrt{2}}{3}&\
\frac{\sqrt{6}}{6}&\frac{3}{32}\\\ \rule{0.0pt}{12.91663pt}\ a&-b&\
\frac{1}{3}&-\frac{1}{3}&\ \frac{1}{3}&\
1&-\frac{\sqrt{2}}{3}&\frac{\sqrt{2}}{3}&-\frac{\sqrt{2}}{3}&\frac{\sqrt{2}}{3}&\
\frac{\sqrt{6}}{6}&\frac{3}{32}\\\ \rule{0.0pt}{12.91663pt}\
b&-a&-\frac{\sqrt{2}}{3}&\frac{\sqrt{2}}{3}&\frac{\sqrt{2}}{3}&-\frac{\sqrt{2}}{3}&\
1&\ \frac{1}{3}&\ \frac{1}{3}&-\frac{1}{3}&-\frac{\sqrt{6}}{6}&\frac{3}{32}\\\
\rule{0.0pt}{12.91663pt}\
b&-a&-\frac{\sqrt{2}}{3}&-\frac{\sqrt{2}}{3}&\frac{\sqrt{2}}{3}&\frac{\sqrt{2}}{3}&\
\frac{1}{3}&\ 1&-\frac{1}{3}&\ \frac{1}{3}&-\frac{\sqrt{6}}{6}&\frac{3}{32}\\\
\rule{0.0pt}{12.91663pt}\
b&-a&\frac{\sqrt{2}}{3}&\frac{\sqrt{2}}{3}&-\frac{\sqrt{2}}{3}&-\frac{\sqrt{2}}{3}&\
\frac{1}{3}&-\frac{1}{3}&\ 1&\ \frac{1}{3}&-\frac{\sqrt{6}}{6}&\frac{3}{32}\\\
\rule{0.0pt}{12.91663pt}\
b&-a&\frac{\sqrt{2}}{3}&-\frac{\sqrt{2}}{3}&-\frac{\sqrt{2}}{3}&\frac{\sqrt{2}}{3}&-\frac{1}{3}&\
\frac{1}{3}&\ \frac{1}{3}&\ 1&-\frac{\sqrt{6}}{6}&\frac{3}{32}\\\
\rule{0.0pt}{12.91663pt}\frac{\sqrt{6}}{6}&\frac{\sqrt{6}}{6}&\frac{\sqrt{6}}{6}&\frac{\sqrt{6}}{6}&\frac{\sqrt{6}}{6}&\frac{\sqrt{6}}{6}&-\frac{\sqrt{6}}{6}&-\frac{\sqrt{6}}{6}&-\frac{\sqrt{6}}{6}&-\frac{\sqrt{6}}{6}&\
1&\frac{1}{10}\\\ \end{array}\right]$

####  5.2.4. $24$ points in $\mathbb{R}^{4}$

The regular $24$ cell, or alternatively the $D_{4}$ root system. The
frequencies of absolute values of inner products are $N(0)=216,\
N(\frac{1}{\sqrt{2}})=144,\ N(\frac{1}{2})=192,\ \text{ and }\ N(1)=24$. The
configuration is unweighted as a projective $3$-design.

####  5.2.5. $16$ points in $\mathbb{R}^{5}$

Lines through antipodal points in the following construction. Take all
permutations of $\pm\frac{1}{\sqrt{30}}(-5,1,1,1,1,1,1)$ and
$\frac{1}{\sqrt{6}}(1,1,1,-1,-1,-1)$ and consider these as vectors in the copy
of $\mathbb{S}^{4}$ in $\mathbb{S}^{5}$ on the plane perpendicular to
$(1,1,1,1,1,1)$. The frequencies of absolute values of inner products are
$N(\frac{1}{3})=90,\ N(\frac{1}{5})=30,\ N(\frac{1}{\sqrt{5}})=120,\ \text{
and }N(1)=16$. The weights making this a projective $2$-design are
$\omega_{1}=\frac{5}{84}$ and $\omega_{2}=\frac{9}{140}$ for the above parts
respectively.

####  5.2.6. $41$ points in $\mathbb{R}^{5}$

An example of a design construction appearing in [Str]. The configuration
comprises of lines through antipodal points in the following construction. Let
$A$ be the set of vectors which are permutations of $(\pm 1,0,0,0,0)$, $B$
permutations of $(\pm\sqrt{\frac{1}{2}},\pm\sqrt{\frac{1}{2}},0,0,0)$, and $C$
permutations of
$(\pm\sqrt{\frac{1}{5}},\pm\sqrt{\frac{1}{5}},\pm\sqrt{\frac{1}{5}},\pm\sqrt{\frac{1}{5}},\pm\sqrt{\frac{1}{5}})$.
The frequencies of absolute values of inner products are $N(0)=600,\
N(\frac{1}{5})=160,\ N(\frac{3}{5})=80,\ N(\sqrt{\frac{1}{5}},\
N(\sqrt{\frac{2}{5}})=320,\ \text{ and }\ N(1)=41$. The weights making this a
projective $3$-design are $\omega_{1}=\frac{2}{105}$,
$\omega_{2}=\frac{8}{315}$, and $\omega_{3}=\frac{25}{1008}$, on $A,B$, and
$C$ respectively.

####  5.2.7. $22$ points in $\mathbb{R}^{6}$

Lines through antipodal points in a hemicube/cross polytope compound, where
the hemicube is within the cube dual to the cross polytope. The frequencies of
absolute values of inner products are $N(0)=30,\ N(\frac{1}{\sqrt{6}})=192,\
N(\frac{1}{3})=240,\ \text{ and }\ N(1)=22$. The weights making this a
projective $2$-design are $\omega_{1}=3/64$ on the hemicube and
$\omega_{2}=1/24$ on the cross-polytope.

####  5.2.8. $63$ points in $\mathbb{R}^{6}$

Lines through antipodal points in the union of minimal vectors of $E_{6}$ and
its dual lattice, $E_{6}^{*}$. The frequencies of absolute values of inner
products are $N(0)=1620,\ N(\frac{1}{4})=432,\ N(\frac{1}{2})=990,\
N(\sqrt{\frac{3}{8}})=864,\ \text{ and }\ N(1)=63$. The weights making this a
projective $3$-design are $\omega_{1}=1/60$ and $\omega_{2}=2/135$ on the
minimal vectors of $E_{6}$ and its dual, respectively.

####  5.2.9. $91$ points in $\mathbb{R}^{7}$

The configuration is projectively composed of the union of the minimal vectors
of $E_{7}$ and its dual lattice, $E_{7}^{*}$. The frequencies of absolute
values of inner products are $N(0)=3906,\ N(\frac{1}{27})=756,\
N(\frac{1}{8})=2016,\ N(\frac{\sqrt{3}}{9})=1512,\ \text{ and }\ N(1)=91$. The
weights making this a projective $3$-design are $\omega_{1}=8/693$ and
$\omega_{2}=3/308$ on the $E_{7}$ part and its dual, respectively. The
cubature formula appears also in [NoS].

####  5.2.10. $36$ points in $\mathbb{R}^{8}$

The edge midpoints of a regular simplex. The frequencies of absolute values of
inner products are $N(\frac{2}{7})=756,\ N(\frac{5}{14})=504,\ \text{ and }\
N(1)=36$. This code is a projective $1$-design with equal weights.

####  5.2.11. $21$ points in $\mathbb{C}^{3}$

A structured union of a maximal simplex (equiangular tight frame, or ETF) of
$9$ vectors and $4$ mutually unbiased bases (a $4$-MUB) of $12$ vectors. The
frequencies of absolute values of inner products are $N(0)=96,\
N(\frac{1}{2})=72,\ N(\frac{1}{\sqrt{3}})=108,\ N(\frac{1}{\sqrt{2}})=144,\
N(1)=21$. The weights making this a projective $3$-design are
$\omega_{1}=4/90$ on the $9$-ETF and $\omega_{2}=\frac{1}{20}$ on the $4$-MUB.

##  6\. $p$-frame energies in non-compact spaces

In the previous sections, we used linear programs to bound energies on compact
two-point homogeneous spaces. This approach can be extended to $p$-frame
energies in non-compact spaces as well. Just as above, we consider
$\mathbb{F}=\mathbb{R},\mathbb{C},$ or $\mathbb{H}$. In this setting, we
consider the set of probability measures $\mathcal{P}(\mathbb{F}^{d})$ with
the additional restriction

(6.1) $\int_{\mathbb{F}^{d}}|x|^{2}d\mu(x)=1$

for each $\mu\in\mathcal{P}(\mathbb{F}^{d})$. This normalization allows us to
obtain a direct extension of above results for the spherical case, and by
scaling, solutions to more general problems can be obtained from these
results. A similar problem of finding maximizers for $p$-frame energies for
$p\leq 2$, subject to the condition that measures be isotropic, was
investigated in [Gl].

For a potential function $f=f(\tau(x,y))$ we define the energy with respect to
measure $\mu\in\mathcal{P}(\mathbb{F}^{d}):$

$I_{f}(\mu)=\int_{\mathbb{F}^{d}}\int_{\mathbb{F}^{d}}f(\tau(x,y))d\mu(x)d\mu(y).$

We will be concerned in this section only with the case that
$f(\tau(x,y))=|\langle x,y\rangle|^{p}$. The Jacobi polynomials for the
projective spaces $\mathbb{FP}^{d-1}$, as above, are denoted $C_{m}$.

######  Lemma 6.1.

For $p\geq 2$, assume $f(t)=\left(\frac{t+1}{2}\right)^{\frac{p}{2}}\geq
h(t)=\sum\limits_{m=0}^{\infty}\widehat{h}_{m}C_{m}(t)$ for all $t\in[-1,1]$,
where $\widehat{h}_{m}\geq 0$ for all $m\geq 0$. Then
$I_{f}(\mu)\geq\widehat{h}_{0}$ for all $\mu\in\mathcal{P}(\mathbb{F}^{d})$
satisfying (6.1).

###### Proof.

Since discrete masses are weak-$*$ dense in $\mathcal{P}(\mathbb{F}^{d})$, it
is sufficient to prove the inequality for them only. Let $\mu$ take the form
$\mu=\frac{1}{N}\sum\limits_{i=1}^{N}\delta_{x_{i}}$, $x_{i}\in\mathbb{F}^{d}$
and set $y_{i}=\frac{x_{i}}{|x_{i}|}$. Then,

$I_{f}(\mu)=\frac{1}{N^{2}}\sum\limits_{i,j=1}^{N}|\langle
x_{i},x_{j}\rangle|^{p}=\frac{1}{N^{2}}\sum\limits_{i,j=1}^{N}|x_{i}|^{p}|x_{j}|^{p}|\langle
y_{i},y_{j}\rangle|^{p}$

$\geq\frac{1}{N^{2}}\sum\limits_{i,j=1}^{N}|x_{i}|^{p}|x_{j}|^{p}h(\tau(y_{i},y_{j}))=\frac{1}{N^{2}}\sum\limits_{m=0}^{\infty}\widehat{h}_{m}\sum\limits_{i,j=1}^{N}|x_{i}|^{p}|x_{j}|^{p}C_{m}(\tau(y_{i},y_{j})).$

For any $m\geq 1$, $C_{m}$ is positive definite on $\mathbb{FP}^{d-1}$ so that
each sum
$\sum\limits_{i,j=1}^{N}|x_{i}|^{p}|x_{j}|^{p}C_{m}(\tau(y_{i},y_{j}))$ is
non-negative. Thus,

$I_{f}(\mu)\geq\widehat{h}_{0}\frac{1}{N^{2}}\sum\limits_{i,j=1}^{N}|x_{i}|^{p}|x_{j}|^{p}C_{0}(\tau(y_{i},y_{j}))=\widehat{h}_{0}\left(\frac{1}{N}\sum\limits_{i=1}^{N}|x_{i}|^{p}\right)^{2}.$

Since $p\geq 2$,

$\frac{1}{N}\sum\limits_{i=1}^{N}|x_{i}|^{p}\geq\left(\frac{1}{N}\sum\limits_{i=1}^{N}|x_{i}|^{2}\right)^{\frac{p}{2}},$

holds by Jensen’s inequality. The constraint 6.1 is equivalent to
$\frac{1}{N}\sum\limits_{i=1}^{N}|x_{i}|^{2}=1$, and so combining all
inequalities, we complete the proof of the lemma. ∎

Lemma 6.1 gives that any linear programming bounds for $p$-frame energies
applicable to the spherical/projective case will work in the non-compact
setting as well. As a consequence of this approach we obtain the following
result.

######  Theorem 6.2.

Let $\mathcal{C}$ be a set of arbitrary unit representatives of a tight
projective $M$-design, $M\geq 2$, in $\mathbb{FP}^{d-1}$ and
$f(\tau(x,y))=|\langle x,y\rangle|^{p}$ with $p\in[2M-2,2M]$. Then

$\mu_{\mathcal{C}}=\frac{1}{|\mathcal{C}|}\sum_{x\in\mathcal{C}}\delta_{x}$

is a minimizer of

$I_{f}(\mu)=\int_{\mathbb{F}^{d}}\int_{\mathbb{F}^{d}}f(\tau(x,y))d\mu(x)d\mu(y)$

over $\mathcal{P}(\mathbb{F}^{d})$, the set of probability measures on
$\mathbb{F}^{d}$ satisfying the constraint 6.1.

###### Proof.

For the proof, we take $f(t)=\left(\frac{t+1}{2}\right)^{\frac{p}{2}}$ and the
interpolating polynomials $H[f,g]$ used in the proof of Theorem 3.7 and follow
the same line of reasoning there:

(6.2) $I_{f}(\mu)\geq I_{H[f,g]}(\mu)\geq
I_{H[f,g]}(\sigma_{d})=I_{H[f,g]}(\mu_{\mathcal{C}})=I_{f}(\mu_{\mathcal{C}}).$

All inequalities are verified in a similiar manner as in the previous section,
except for $I_{H[f,g]}(\mu)\geq I_{H[f,g]}(\sigma_{d})$. This part follows
from Lemma 6.1 applied to $h=H[f,g]$ because $I_{h}(\sigma_{d})$ is precisely
$\widehat{h}_{0}$ for positive definite functions $h$. ∎

To conclude the section, we note that the results analogous to Theorem 4.2 on
the optimality of the 600-cell and the numerical linear programming bounds for
$p$-frame energies in the compact setting apply to the non-compact setting
also.

##  7\. Mixed volume inequalities

In this section we demonstrate an intriguing connection between the $p$-frame
energy and convex geometry. We begin by briefly recalling some of the basic
notions from convex geometry. See [K, Ch. 2] for a more thorough development.

Let $K$ be a convex body and $\sigma_{K}(u)$ be the surface measure of $K$,
that is, a measure supported on the unit sphere $\mathbb{S}^{d-1}$, satisfying

$\sigma_{K}(B)=|x\in\partial K,\ \text{the outer unit normal to }K\ \text{at}\
x\ \text{belongs to }B|_{d-1}$

for all Borel sets $B\subset\mathbb{S}^{d-1}$, where $|\cdot|_{d-1}$ denotes
the $(d-1)$-dimensional Hausdorff measure. For example, if $K$ is a polytope
with faces $\\{K_{i}\\}_{i=1}^{m}$ and normals $\\{{n}_{i}\\}_{i=1}^{m}$,
$\sigma_{K}$ is atomic with mass $|K_{i}|_{d-1}$ at each ${n}_{i}$,

$\sigma_{K}=\sum\limits_{i=1}^{m}|K_{i}|_{d-1}\delta_{{n}_{i}},$

and if $K=\mathbb{B}$ is the $d$-dimensional unit ball, then $\sigma_{K}$
simply coincides with the standard (unnormalized) uniform surface area measure
$\sigma_{K}(B)=|B|_{d-1}=\frac{2\pi^{d/2}}{\Gamma(d/2)}\sigma(B)$.

Recall that for a convex body, $K\subset\mathbb{R}^{d}$, the support function
$h_{K}(u)$ of $K$ takes the form

$h_{k}(u)=\sup\limits_{v\in K}\langle u,v\rangle.$

Given two convex bodies $K$ and $L$, and $p\geq 1$, define

$V_{p}(K,L)=\frac{p}{d}\lim\limits_{\epsilon\rightarrow
0}\frac{|K+_{p}\epsilon L|-|K|}{\epsilon},$

where $K+_{p}\epsilon L$ is the convex body with support function
$h_{K+_{p}\epsilon L}(u)$ satisfying

$h_{K+_{p}\epsilon L}(u)^{p}=h_{K}(u)^{p}+\epsilon h_{L}(u)^{p}.$

Note that for $L=\mathbb{B}_{d}$ is the unit ball and $p=1$, the above
quantity is just the definition of the surface area of $K$. In general,
$V_{p}(K,L)$ is known as the $L_{p}$-mixed volume of $K$ and $L$. The
following alternative integral representation for $V_{p}(K,L)$ is known

$V_{p}(K,L)=\frac{1}{d}\int_{\mathbb{S}^{d-1}}h_{L}(u)^{p}d\sigma_{K}^{p}(u),$

where $d\sigma_{K}^{p}(u)=h_{K}(u)^{1-p}d\sigma_{K}(u)$, so that in particular
$d\sigma_{K}^{1}(u)=d\sigma_{K}(u)$ .

Now, call a probability measure $\mu$ supported on $\mathbb{S}^{d-1}$
admissible, if it is symmetric and not concentrated on a subspace. A classical
result which follows from Minkowski’s theorem, says that any admissible
measure can be realized as the surface area measure of a symmetric convex
body; see more in [Sch, Ch. 7].

The projection body $\Pi K$ of a convex body $K$ is defined to be a body such
that for each $u\in\mathbb{S}^{d-1}$

$h_{\Pi K}(u)=\left|K|u^{\perp}\right|_{d-1},$

that is, the support function of $\Pi K$ equals the volume of the projection
of $K$ onto the hyperplane orthogonal to $u$ [BL]. Since

$\left|K|u^{\perp}\right|_{d-1}=\frac{1}{2}\int_{\mathbb{S}^{d-1}}|\langle
u,v\rangle|d\sigma_{K}(v),$

the identities

$\displaystyle I_{|t|}(\sigma_{K})$
$\displaystyle=\int_{\mathbb{S}^{d-1}}\int_{\mathbb{S}^{d-1}}|\langle
u,v\rangle|\,d\sigma_{K}(u)d\sigma_{K}(v)=2\int_{\mathbb{S}^{d-1}}\left|K|u^{\perp}\right|_{d-1}d\sigma_{K}(u)$
$\displaystyle=2\int_{\mathbb{S}^{d-1}}h_{\Pi
K}(u)\,d\sigma_{K}(u)=2d\,V_{1}(K,\Pi K)$

finally establish the connection between $L_{1}$-mixed volumes and $1$-frame
energies.

Our main theorem, Theorem 1.1, shows that all minimizers of $I_{|t|^{p}}(\mu)$
over probability measures are admissible when a corresponding tight design
exists, as this measure is both discrete and can be taken to be symmetric.
From this, we obtain what appears to be a new observation, namely the
following:

######  Proposition 7.1.

The minimum of the quantity

$\frac{V_{1}(K,\Pi K)}{|\partial K|^{2}}$

over all symmetric convex bodies in $\mathbb{R}^{d}$ is achieved when $K$ is a
cube.

Indeed, it is easy to see that, when $K$ is a cube, the surface measure
$\sigma_{K}$ is equally distributed on the vertices of a cross-polytope, which
minimizes the $p$-frame energy for $p=1$.

One may also define $L^{p}$-intersection bodies $\Pi_{p}K$ [LYZ] in a similar
fashion and obtain analogous relations for other values of $p$. Doing so
allows one to infer similar statements for $V_{p}(K,\Pi_{p}K)/|\partial
K|^{2}$ for the several dimensions and ranges of $p$ considered in this
manuscript (for which tight designs exist), as well as pose conjectures
corresponding to the numerically obtained minimizers. We anticipate, in
particular, in accordance with Conjecture 1.3, that whenever $p$ is not an
even integer, this quantity is always minimized by a convex body which is
polyhedral (with discrete surface measure).

##  8\. Other linear programming applications

###  8.1. Other energy problems

We now discuss some problems related to minimization of $p$-frame energies and
other energies with degree $M$ absolutely monotonic potentials. For $p\not\in
2\mathbb{N}$, the potential functions
$f(t)=\frac{1}{2^{p/2}}\cdot(1+t)^{p/2}$, corresponding to the $p$-frame
energy, have the property that not only their derivatives switch signs for
large enough orders, but also the coefficients in their Jacobi expansion have
alternating signs. While the proof of optimality depended heavily on the
former, we look into the latter property now.

In a sense, the most natural polynomial potential functions to consider when
approximating $f(t)$ are of the form

(8.1) $g(t)=\sum\limits_{m=0}^{k}\widehat{f}_{m}C_{m}(t)-\beta C_{k+1}(t),$

where $\widehat{f}_{m},\beta\geq 0$. Because any earlier truncation of
function $f$ is positive definite, so that $I_{f}(\mu)$ is minimized by
surface measure, the point at which the first negative coefficient comes in is
the first interesting truncation to consider. At first glance, it may seem
that minimizers of $I_{g}(\mu)$ might act like those of $I_{f}(\mu)$. When
$\beta$ is too large however, this cannot be true, for in this case a single
Dirac mass $\nu=\delta_{x}$ gives

$I_{g}(\nu)=g(\tau(x,x))=\sum\limits_{m=0}^{k}\widehat{f}_{m}-\beta,$

which can be smaller than the value obtained on any other measure. Instead of
looking at the energy with potential $g$, we shall considering the limiting
problem of constrained optimization

(8.2)
$\max\limits_{\mu\in\mathcal{P}(\Omega)}\int_{\Omega}\int_{\Omega}C_{k+1}(\tau(x,y))d\mu(x)d\mu(y)\
\text{s.t.}\ \int_{\Omega}\int_{\Omega}C_{j}(\tau(x,y))d\mu(x)d\mu(y)=0,\
j=1,...,k,$

which is again minimized by tight designs, as the below argument shows.

######  Theorem 8.1.

If a tight $k$-design $\mathcal{C}$ in $\Omega$ exists, then the measure
$\mu_{\mathcal{C}}$ which distributes mass evenly among the points of
$\mathcal{C}$ solves the optimization problem (8.2) over probability measures.

###### Proof.

Set $f=C_{k+1}$. If $k=2m$, we construct the polynomial $h$ by applying
Hermite interpolation to $f$ at $t=1$, and to $f$ and $f^{\prime}$ at the
other $m$ values of
$\mathcal{A}(\mathcal{C})=\\{\tau(x,y)|\,x,y\in\mathcal{C},x\neq y\\}$, so
that $\textup{deg}\,h\leq 2m=k$. When $k=2m-1$, i.e. $\mathcal{C}$ contains
antipodal pairs, we apply interpolation of order $1$ at $t=\pm 1$ and of order
$2$ at the other $m-1$ values of $\mathcal{A}(\mathcal{C})$, resulting in
$\textup{deg}\,h\leq 1+2(m-1)=2m-1=k$. The remainder formula (3.1) then gives
that the difference

$f(t)-h(t)=\frac{f^{(k+1)}(\xi)}{(k+1)!}(t-1)\cdot\begin{cases}\prod\limits_{\alpha\in\mathcal{A}(\mathcal{C})\setminus\\{1\\}}(t-\alpha)^{2}\\\
\prod\limits_{\alpha\in\mathcal{A}(\mathcal{C})\setminus\\{\pm
1\\}}(t-\alpha)^{2}\cdot(t+1)\end{cases}$

is non-positive for $t\in[-1,1]$. This holds because $f^{(k+1)}(\xi)>0$, as it
is simply the leading coefficient of $f=C_{k+1}$. Since $h$ is a polynomial of
degree $k$, the constraints in (8.2) imply that for any admissible
$\mu\in\mathcal{P}(\Omega)$, $I_{h}(\mu)=\widehat{h}_{0}=I_{h}(\sigma).$ We
therefore obtain

$I_{f}(\mu)\leq
I_{h}(\mu)=I_{h}(\sigma)=I_{h}(\mu_{\mathcal{C}})=I_{f}(\mu_{\mathcal{C}}),$

where the penultimate equality relies on the fact that $\mathcal{C}$ is a
$k$-design, and the last one follows from interpolation. ∎

Note that the argument for uniqueness applies for the above problem, much as
it did in the case of degree $M$ absolutely monotonic functions $f$. The
difference lies in the fact that here the design condition arises from the
constraints.

It is possible to show that generally, both for problem (8.2) above and the
problem of minimizing $g$-energy for $g$ as in (8.1), there exist finitely
supported minimizing measures. The reader familiar with the geometry of moment
constrained probability measures will not be surprised by this since the
extreme points of sets of such measures are finitely supported. The complete
argument for existence of discrete measures can be found in [BGM+], but a
quick sketch is as follows. Each problem above has solutions which may be
found by re-writing the problem as maximization of a convex functional over
the set of moment constrained probability measures (the second problem (8.2)
is already exactly in such form) and thus the maximum should be attained on an
extreme point, i.e. a discrete measure. In both cases, in addition to the
existence of a discrete minimizer, one can obtain quantitative upper bounds on
the support size in terms of the number of constraints.

###  8.2. Causal variational principle

Define the kernel

(8.3)
$\mathcal{L}(t)=\mathcal{L}_{\tau}(t):=\max\\{0,2\tau^{2}\big{(}1+t\big{)}\big{(}2-\tau^{2}(1-t)\big{)}\\}.$

for $\tau>0$. The minimization problem for the energy

(8.4)
$I_{\mathcal{L}}(\mu)=\int_{\mathbb{S}^{2}}\int_{\mathbb{S}^{2}}\mathcal{L}(x\cdot
y)d\mu(x)d\mu(y)$

is known as the causal variational principle on the sphere and is connected to
relativistic quantum field theory. It is conjectured in [FS] that all the
minimizers of (8.4) are discrete whenever $\tau>\sqrt{2}$ based on numerical
evidence (it is also conjectured that there exist discrete minimizers for
$\tau\geq 1$).

Here we confirm this conjecture for two values of $\tau>0$, for which we can
show that the cross-polytope or orthoplex and the icosahedron indeed minimize
the energy. These minimizing measures were suggested by numerical experiments
in [FS].

####  8.2.1. Cross-polytope: $\tau=\sqrt{2}$

When $\tau=\sqrt{2}$, we have

$\mathcal{L}(t)=\max\\{0,8t^{2}+8t\\},$

and thus $\mathcal{L}(0)=0$. Define the measure

$\nu=\frac{1}{6}\sum_{i=1}^{3}\bigg{(}\delta_{e_{i}}+\delta_{-e_{i}}\bigg{)},$

where $\\{e_{1},e_{2},e_{3}\\}$ is an orthonormal basis of $\mathbb{R}^{3}$,
i.e. $d\nu$ is a measure whose mass is equally concentrated in the vertices of
a cross-polytope. Then we have,

######  Proposition 8.2.

The measure $\nu$ is a minimizer for the energy, $I_{\mathcal{L}}$, over
probability measures on $\mathbb{S}^{2}$ for $\tau=\sqrt{2}$.

####  8.2.2. Icosahedron:
$\displaystyle{\tau^{2}=\frac{2\sqrt{5}}{\sqrt{5}-1}}$

This value of $\tau$ is chosen so that $\mathcal{L}_{\tau}(1/\sqrt{5})=0$. Let
$\mathcal{C}\subset\mathbb{S}^{2}$ be the vertices of a regular icosahedron
and let

$\displaystyle{\nu=\frac{1}{12}\sum_{x\in\mathcal{C}}\delta_{x}}$

be the uniform measure on the vertices of the icosahedron.

######  Proposition 8.3.

The measure $\nu$ is a minimizer for the energy, $I_{\mathcal{L}}$, over
probability measures on $\mathbb{S}^{2}$ for
$\tau^{2}=\frac{2\sqrt{5}}{\sqrt{5}-1}$.

The proof of Propositions 8.2 and 8.3 is another application of the linear
programming framework, which in this case is particularly straightforward,
since, unlike the previous sections, a single auxiliary function must be
constructed to certify the solution. We postpone the details to the Appendix,
see Section 11.3.

##  9\. Further remarks

We have many remaining questions about the $p$-frame energies, and many
curiosities were brought to our attention through our numerical study. One
immediate question concerns uniqueness of the $600$-cell as a minimizer for
$\mathbb{RP}^{3}$ and $p\in(8,10)$, which we expect to hold. In Section 3.5,
we mentioned that tight designs, generally, are not unique (not even up to
unitary equivalence). This is known to be true in particular for SIC-POVM’s in
$\mathbb{C}^{3}$ in [Sző]. It is interesting whether it is more often the case
that this happens or that such configurations are isolated, as is known to be
the case when $d=2$ [Z].

An interesting observation is that some configurations minimize $p$-frame
energies for a range of $p$ (the $600$-cell for example), while others, like
the $p=3$ minimizer in $\mathbb{RP}^{7}$, do not minimize on an entire range
between even integers. When minimizers have the same support for a range
$p\in(2k-2,2k)$, it indicates that the supporting configuration has to be a
weighted $k$-design.

For the $36$ points in $\mathbb{RP}^{7}$ given as the midpoints of edges of a
regular simplex, one can check that the strength of this configuration as a
design is too small to satisfy the above condition. Further, the value of the
energy for a measure which equally distributes over this set when compared
against the surface measure is too large to be a minimizer when $p$ is close
to (but less) than four.

We do not expect that the minimizers of $p$-frame energies are necessarily
weighted $k$-designs, but noticed that many of the configurations which showed
up numerically as limit points of the even $p$ values (from below) were
smallest known weighted designs. Informally, one might expect for these
configurations to have isolated or small support since if the points become
too well distributed, the distribution gets “closer” to surface measure which
means the averages of the configuration over the negative coefficient terms in
the Jacobi expansion of $f$ vanish. Since one wants to maximize such
contributions, the vectors might be taken close to a weighted $k$-design, but
just “barely” so.

Some other cases where the support of a minimizer appears to change within
even arguments are for $\mathbb{RP}^{2}$, $p\in(4,6)$ and $\mathbb{RP}^{3}$,
$p\in(2,4)$. One might be tempted to suggest that the configurations which
show up as minimizers on an interval are universally optimal, but this is not
the case. For example, the $D_{4}$ root system, which appears to be optimal on
$p\in(4,6)$ for $\mathbb{S}^{3}$, is not universally optimal [CCE+].
Nonetheless, in the limited numerical experiments which were run outside of
the parameters found in Tables 1 and 2, it appears uncommon that a
configuration be optimal on a range of $p$, and when it does happen, the
configuration is highly structured.

This suggests another phenomenon similar to the notion of universal
optimality, and we are tempted to conjecture that in the real case for $d>2$
there are only finitely many configurations which optimize the $p$-frame
energy on a whole range of $p\in[2k-2,2k]$.

As was mentioned earlier, we conjecture that all energies with $p$ not even
have discrete and only discrete minimizers. It is known already that $p$-frame
energies among other non-positive definite functions cannot contain an open
set in the interior of the support of a minimizing measure [BGM+]. We are not
familiar with an argument which would rule out the possibility of an arc of a
circle being contained in the support of a minimizing measure for $p$-frame
potentials, or non-positive definite truncations of their Jacobi polynomial
expansions.

Looking at the tables, one can note that as the value of $p$ increases, for
$p$ not even, the support size of a candidate appears to be monotonically
increasing. Further, for a fixed dimension, the support size seems to grow
polynomially in $p$. We do not have an explanation for this phenomenon.

One motivating reason for considering the other projective spaces beyond the
real case, is the connection with the problem known as Zauner’s conjecture, on
existence of tight projective $2$-designs, these being best known by their
alternative name SIC-POVMs. The moment constrained problem considered in
Section 8 for $k=2$ has the property that a discrete solution with support
size bounded by an explicit function of $d$ exists regardless of whether a
tight design exists. Further, the minimizer must be a (weighted) projective
$2$-design. If a SIC-POVM exists, it must solve this problem or the $p$-frame
energy problem for $p\in(2,4)$.

Interestingly, it is conjectured [CKM] from numerical evidence that the
property analogous to Zauner’s should not hold in the quaternionic setting. If
this is true, it is curious what instead should appear as a minimizer.

Finally, we give additional details on how we made the conjectures found in
Tables 1 and 2. The numerical method employed to find conjectured minimizers
involved two steps. Early on, we used a quick first order gradient descent
method to minimize energies. Afterwards we implemented an arbitrary precision
library with a second order method to check our conjectures and test endpoint
behavior.

##  10\. Acknowledgements

We express our gratitude to the following organizations that hosted subsets of
the authors during the work on this paper: AIM, ICERM, INI, CIEM, Georgia
Tech. The first author was supported by the grant DMS-1665007, the third
author was supported by the Graduate Fellowship 00039202, and the fourth
author was supported in part by the grant DMS-1600693, all from the US
National Science Foundation. This work is supported in part by EPSRC grant no
EP/K032208/1. We are extremely grateful to Galyna Livshyts for bringing to our
attention the alternative interpretation of the $p$-frame energy, as given in
Section 7.

We thank Greg Blekherman, Henry Cohn, David de Laat, Michael Lacey, Bruce
Reznick, Alexander Reznikov, Daniela Schiefeneder, and Yao Yao for helpful
discussions.

##  11\. Appendix

###  11.1. Parameters of the conjectured and rigorous optimizing
configurations

Table 7 gives weights and inner products of the support vectors of $p$-frame
minimizing measures in $\mathbb{R}^{d}$, see Table 1 and Section 5.2.

Table 7. Optimal and conjectured configurations for $p$-frame energies. The
configurations are supported on $N$ unit norm vectors in $\mathbb{R}^{d}$, and
are strength $M$ designs. Only the absolute values of inner products are
given.
Note: $\alpha=\frac{\sqrt{75+30+\sqrt{5}}}{15},\beta=\sqrt{\frac{1}{15}(5-2\sqrt{5})},\text{ and }\gamma=\frac{\sqrt{6-2\sqrt{5}}}{6}$. $d$ | $N$ | $M$ | Weights | Inner Products | Name
---|---|---|---|---|---
$2$ | $N$ | $N-1$ | ${1}/{N}$ | $|\cos(2\pi j/N)|,1\leq j<N$ |  $N$-gon
$d$ | $d$ | $1$ | ${1}/{N}$ | $0$ | cross polytope
$3$ | $6$ | $2$ | ${1}/{N}$ | $\frac{1}{\sqrt{5}}$ | icosahedron
$3$ | $11$ | $3$ | $\frac{1}{10},\frac{2}{27},\frac{49}{540}$ | $0,\frac{1}{7},\frac{4}{7},\frac{5}{7},\sqrt{\frac{1}{7}},\sqrt{\frac{3}{7}},\sqrt{\frac{4}{7}}$ | Reznick weighted design
$3$ | $16$ | $4$ | $\frac{5}{84},\frac{9}{140}$ | $\frac{1}{3},\frac{1}{\sqrt{5}},\sqrt{\frac{5}{9}},\alpha,\beta$ | icosahedron and dodecahedron
$4$ | $11$ | $2$ | $\frac{1}{10},\frac{3}{32},\frac{3}{40}$ | $\frac{1}{3},\frac{\sqrt{2}}{3},\frac{\sqrt{6}}{6},\frac{\sqrt{5}+1}{6},\gamma$ | weighted design
$4$ | $24$ | $3$ | ${1}/{N}$ | $0,{1}/{2},{1}/{\sqrt{2}}$ |  $D_{4}$ roots
$4$ | $60$ | $5$ | ${1}/{N}$ | $0,\frac{\sqrt{5}\pm 1}{4},\frac{1}{2}$ |  $600$-cell
$5$ | $16$ | $2$ | $\frac{5}{84},\frac{9}{140}$ | $\frac{1}{5},\frac{1}{3},\frac{1}{\sqrt{5}}$ | hemicube
$5$ | $41$ | $3$ | $\frac{25}{1008},\frac{8}{315},\frac{2}{105}$ | $0,\frac{1}{5},\frac{1}{2},\frac{3}{5},\frac{1}{\sqrt{2}},\frac{1}{\sqrt{5}},\sqrt{\frac{2}{5}}$ | Stroud weighted design
$6$ | $22$ | $2$ | $\frac{3}{64},\frac{1}{24}$ | $0,\frac{1}{3},\frac{1}{\sqrt{6}}$ | cross polytope and hemicube
$6$ | $63$ | $3$ | $\frac{2}{135},\frac{1}{60}$ | $0,\frac{1}{4},\frac{1}{2},\sqrt{\frac{3}{8}}$ |  $E_{6}$ and $E_{6}^{*}$ roots
$7$ | $28$ | $2$ | $\frac{1}{28}$ | $1/3$ | kissing $E_{8}$
$7$ | $91$ | $3$ | $\frac{3}{308},\frac{8}{693}$ | $0,\frac{1}{27},\frac{1}{8},\frac{\sqrt{3}}{9}$ |  $E_{7}$ and $E_{7}^{*}$ roots
$8$ | $36$ | $1$ | ${1}/{N}$ | ${5}/{14},{2}/{7}$ | mid-edges of regular simplex
$8$ | $120$ | $3$ | ${1}/{N}$ | $0,1/2$ |  $E_{8}$ roots
$23$ | $276$ | $2$ | ${1}/{N}$ | $1/5$ | equiangular
$23$ | $2300$ | $3$ | ${1}/{N}$ | $0,1/3$ | kissing $\Lambda_{24}$
$24$ | $98280$ | $5$ | ${1}/{N}$ | $0,1/4,1/2$ |  $\Lambda_{24}$, Leech lattice roots

###  11.2. Numerical LP bounds

Table 8 collects linear programming lower bounds corresponding to small values
of $d$ and odd values $p$ for the $p$-frame energy on $\mathbb{S}^{d-1}$.

Table 8. Numeric linear programming lower bounds for odd-valued $p$-frame energies. $d$ | $p=3$ | $p=5$ | $p=7$
---|---|---|---
$3$ | $0.2412$ | $0.1655$ | $0.1248$
$4$ | $0.1612$ | $0.09607$ | $0.06454$
$5$ | $0.1170$ | $0.06169$ | $0.03740$
$6$ | $0.08970$ | $0.04240$ | $0.02344$
$7$ | $0.07142$ | $0.03060$ | $0.01556$
$8$ | $0.05852$ | $0.02291$ | $0.01080$
$9$ | $0.04902$ | $0.01770$ | $0.007768$
$10$ | $0.04180$ | $0.01401$ | $0.005750$
$11$ | $0.03616$ | $0.01131$ | $0.004360$
$12$ | $0.03166$ | $0.009290$ | $0.003375$
$13$ | $0.02801$ | $0.007737$ | $0.002658$
$14$ | $0.02499$ | $0.006524$ | $0.002125$
$15$ | $0.02248$ | $0.005561$ | $0.001721$
$16$ | $0.02035$ | $0.004785$ | $0.001413$
$17$ | $0.01853$ | $0.004152$ | $0.001171$
$18$ | $0.01696$ | $0.003630$ | $0.0009813$
$19$ | $0.01559$ | $0.003195$ | $0.0008280$
$20$ | $0.01440$ | $0.002830$ | $0.0007054$
$21$ | $0.01335$ | $0.002520$ | $0.0006047$
$22$ | $0.01242$ | $0.002256$ | $0.0005217$
$23$ | $0.01159$ | $0.002028$ | $0.0004529$
$24$ | $0.01085$ | $0.001832$ | $0.0003952$

###  11.3. Causal variational principle

####  11.3.1. Cross-polytope

Let the following polynomial be given,

$H(t)=8t^{2}+8t.$

It is easy to see that $H$ is positive definite on $\mathbb{S}^{2}$.
Additionally, it is obvious that

$H(t)\leq\mathcal{L}(t)\qquad\textup{ for all }t\in[-1,1],$

and

$H(-1)=\mathcal{L}(-1)=0,\quad H(0)=\mathcal{L}(0)=0,\quad
H(1)=\mathcal{L}(1)=16,$

so that $H$ coincides with $\mathcal{L}$ on the set $\\{x\cdot
y:x,y\in\operatorname{supp}\nu\\}$.

We obtain that for any measure $\mu\in\mathcal{P}$,

(11.1) $I_{\mathcal{L}}(\mu)\geq I_{H}(\mu)\geq
I_{H}(\sigma)=I_{H}(\nu)=I_{\mathcal{L}}(\nu),$

where we have used the fact that $H(t)\leq\mathcal{L}(t)$ for $t\in[-1,1]$, so
$I_{\mathcal{L}}(\mu)\geq I_{H}(\mu)$. Since $H$ is positive definite,
according to Propositions 2.2 and 2.3, we have that $\sigma$ minimizes
$I_{H}$, i.e. $I_{H}(\mu)\geq I_{H}(\sigma)$. We have also used that the
cross-polytope is a $3$-design and $H$ is a quadratic polynomial, hence
$I_{H}(\sigma)=I_{H}(\nu)$. Finally, $H(t)=\mathcal{L}(t)$ for $t\in\\{x\cdot
y:x,y\in\operatorname{supp}\nu\\}=\\{0,\pm 1\\}$, hence
$I_{H}(\nu)=I_{\mathcal{L}}(\nu)$. This proves that the cross-polytope
minimizes $I_{\mathcal{L}}$ for $\tau=\sqrt{2}$.

####  11.3.2. Icosahedron

We shall need two facts about the icosahedron, namely that the set of inner
products between elements of $\mathcal{C}$ is $\\{x\cdot
y:x,y\in\mathcal{C}\\}=\left\\{\pm 1,\pm 1/{\sqrt{5}}\right\\}$, and that the
icosahedron $\mathcal{C}$ is a $5$-design. For simplicity let us consider the
function $F(t)=\frac{\mathcal{L}(t)}{\mathcal{L}(1)}$ so that $F(1)=1$
(obviously, this does not effect the minimizers).

We construct the following polynomial:

$\displaystyle H(t)$
$\displaystyle=\frac{5(5-\sqrt{5})}{32}t^{4}+\frac{5}{8}t^{3}+\frac{3\sqrt{5}-5}{16}t^{2}-\frac{1}{8}t+\frac{1-\sqrt{5}}{32}$
$\displaystyle=\frac{5-\sqrt{5}}{28}C_{4}(t)+\frac{1}{4}C_{3}(t)+\frac{20+3\sqrt{5}}{84}C_{2}(t)+\frac{1}{4}C_{1}(t)+\frac{1}{12}C_{0}(t),$

where $C_{k}$ are the standard Legendre polynomials. We observe then that $H$
is positive definite, and that $H(t)\leq F(t)$ for $-1\leq t\leq 1$, which
follows from the formula

$H(t)=\frac{5}{32}(5-\sqrt{5})(t+1)(t-\frac{1}{\sqrt{5}})(t+\frac{1}{\sqrt{5}}).$

A glance at this formula gives $H\leq F$ for $t\in[-1,\frac{1}{\sqrt{5}}]$,
and the fact that $F-H$ is a polynomial with roots

$t=-1,\frac{1}{\sqrt{5}},\frac{-1\pm 4\sqrt{10+4\sqrt{5}}}{\sqrt{5}},$

gives $H\leq F$ for $t\in[\frac{1}{\sqrt{5}},1]$ which is a subset of the
interval $[\frac{1}{\sqrt{5}},\frac{-1+4\sqrt{10+4\sqrt{5}}}{\sqrt{5}}]$.

$H$ coincides with $F$ (by construction) on the set $\\{\langle
x,y\rangle:x,y\in\mathcal{C}\\}=\left\\{\pm 1,\pm 1{\sqrt{5}}\right\\}$. $H$
has a local maximum at $-\frac{1}{\sqrt{5}}$ and has been obtained by solving
the linear equations $H(t)=F(t)$ for $t=\pm 1,\pm 1/{\sqrt{5}}$, as well as
$H^{\prime}(-1/\sqrt{5})=0$. The same argument as in the previous subsection
finally shows

$\displaystyle{I_{F}(\nu)=\inf_{\mu\in\mathcal{P}}I_{F}(\mu)},$

i.e. the icosahedron minimizes the energy $I_{\mathcal{L}}$ for
$\tau^{2}=\frac{2\sqrt{5}}{\sqrt{5}-1}$.

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