# Energy on spheres and discreteness of minimizing measures

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>
###### Key words and phrases:

Potential energy minimization, spherical codes, spherical designs, attractive-
repulsive

###### 2010 Mathematics Subject Classification:

Primary 52A40, 31E05; Secondary 58C35, 90C26

##  1\. Introduction

Energy minimization on the sphere arises naturally in numerous contexts in
mathematical physics, discrete and metric geometry, coding theory, signal
processing, and other fields of mathematics. Many problems can be reformulated
in terms of minimization of the discrete energy

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

over all $N$-point configurations $\mathcal{C}\subset\mathbb{S}^{d-1}$, or of
the continuous energy integral

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

over $\mu\in\mathcal{P}(\mathbb{S}^{d-1})$, the set of all Borel probability
measures. In this work, we mostly concentrate on the energy integrals. We
assume that the measurable function $f:[-1,1]\rightarrow\mathbb{R}$ is bounded
below, hence the integral (1.2) is well defined, although it may be infinite
for some measures.

Loosely speaking, minimizing the discrete $N$-point energy may be interpreted
as finding the equilibrium position of $N$ “particles” on the sphere, which
interact according to the potential $f$, which depends on the distance between
$x$ and $y$, while minimizing the energy integral corresponds to finding the
optimal distribution of unit charge on $\mathbb{S}^{d-1}$ under the same
interaction. Minimization of energy integrals dictates the limiting behavior
of the discrete problem as the number of points $N$ goes to infinity. Observe
that the interaction depends only on the distance between $x$ and $y$, hence
the energy (1.2) is invariant under orthogonal transformations.

The definitions of the discrete (1.1) and continuous (1.2) energies are
compatible in the sense that

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

and we shall often abuse the terminology by saying that $\mathcal{C}$ (instead
of $\mu_{\mathcal{C}}$) minimizes $I_{f}$.

In some models, energy minimization leads to a clustering effect, in the sense
that the resulting optimal measures tend to be discrete or at least supported
on lower dimensional submanifolds. This phenomenon has been repeatedly
observed for energies on $\mathbb{R}^{d}$ with attractive-repulsive
potentials, which naturally appear in models in computational chemistry,
mathematical biology, and social sciences [BCL+, CFP, CMV, KKLS, KSU+, MEB+,
VUK+, WS].

In many instances, it is observed that in the Euclidean setting the above
energies are minimized by measures supported on a sphere of some radius. Our
results have some implications in this direction, but we concentrate primarily
on attractive-repulsive potentials on the sphere, i.e. functions $f(\langle
x,y\rangle)$ which are increasing near $1$, but decreasing near $-1$; in other
words, two particles $x$ and $y$ experience repulsion when $x$ and $y$ are
close, but attract when they are far apart. In some examples, potentials of
the energy are also symmetric and orthogonalizing, i.e. they satisfy
$f(t)=f(|t|)$, and $\min\\{f(t):\,t\in[-1,1]\\}=f(0)$, which results in two
particles achieving equilibrium when they are in an orthogonal position.

One of the most interesting energies of this type is the $p$-frame energy
corresponding to $f(t)=|t|^{p}$, where $p>0$,

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

The behavior of minimizing measures of this energy exhibits peculiar phase
transitions at even integer values of $p$. Whenever $p\in 2\mathbb{N}$, the
$p$-frame energy is minimized by the normalized surface measure $\sigma$ [Si,
EO], among other measures. However, for $p\not\in 2\mathbb{N}$, all the
minimizers appear to be discrete [BGM+].

For $p=2$, this energy and its discrete counterpart, often referred to simply
as the frame potential, have been studied in [Si] and later again in [BeF]. In
the latter paper, which coined the name for this energy, it was proved that
the minimizers of the discrete energies with $N\geq d$ points are precisely
unit norm tight frames. A tight frame is a set of vectors
$\\{\varphi_{i}\\}_{i=1}^{N}\subset\mathbb{R}^{d}$ such that a Parseval type
identity,

$\sum\limits_{i=1}^{N}|\langle x,\varphi_{i}\rangle|^{2}=A\|x\|^{2},$

holds for all $x\in\mathbb{R}^{d}$ and for some constant $A>0$. In other
words, tight frames act as overcomplete orthonormal bases and thus play an
important role in several areas of applied mathematics. It is easy to see that
tight frames, and more generally, isotropic measures on the sphere also
minimize the continuous frame energy over all probability measures.

In the case $p=4$, the $p$-frame energy is closely connected to the maximal
equiangular tight frames, which in the complex case are known as symmetric
informationally complete positive operator-valued measures (SIC-POVMs). These
are unit norm tight frames $\\{\varphi_{i}\\}_{i=1}^{N}$ with the property
that $|\langle\varphi_{i},\varphi_{j}\rangle|^{2}=\frac{1}{d+2}$ or
$\frac{1}{d+1}$ for $i\neq j$, in the real and complex case respectively. In
$\mathbb{C}^{d}$, Zauner’s conjecture [Z] states that SIC-POVMs exist in all
dimensions $d\geq 2$, which is supported by extensive numerical evidence [SG,
RBSC]. In the real case, the existence of analogous objects is also
mysterious: they may exist only in dimensions $d=(2m-1)^{2}-2$ [LS], but do
not exist for $d=47$ [Ma]. When these objects do exist, they minimize the
$4$-frame energy (with the complex unit sphere replacing $\mathbb{S}^{d-1}$ in
the case of SIC-POVMs).

More generally, when $p=2k\in 2\mathbb{N}$, the function $f(t)=|t|^{p}=t^{2k}$
is a polynomial, hence any spherical $2k$-design yields the same value of the
$p$-frame energy as $\sigma$, and thus is also a minimizer. More precisely,
discrete equal-weight minimizers are exactly projective $k$-designs. A
spherical $t$-design is a set $\\{x_{i}\\}_{i=1}^{N}\subset\mathbb{S}^{d-1}$
such that

$\frac{1}{N}\sum_{i=1}^{N}P(x_{i})=\int_{\mathbb{S}^{d-1}}P(x)d\sigma(x)$

for any polynomial $P$ on $\mathbb{R}^{d}$ of degree up to $t$; see e.g.
[DGS], while a projective $k$-design is a configuration such that the above
identity holds for all polynomials of degree up to $2k$, which contain only
even-degree terms. To summarize, the $p$-frame energy has a multitude of
minimizers, both continuous and discrete, when $p$ is an even integer.

When $p\not\in 2\mathbb{N}$, the situation is much less studied. In our recent
paper [BGM+], we have shown that, when certain highly symmetrical
configurations exist, they minimize the $p$-frame energy on a range of values
of $p$ between two consecutive even integers. These configurations, known as
tight designs, are designs of high order with few distinct pairwise distances,
or equivalently, designs of smallest possible cardinality [DGS]. Theorem 1.1
in [BGM+] states that a tight spherical $(2k+1)$-design, whenever it exists,
minimizes the $p$-frame energy for $p\in[2k-2,2k]$, and, moreover, every
minimizer for $p\in(2k-2,2k)$ has to be a tight design (in particular, it has
to be discrete). In addition, we have accumulated a great deal of numerical
evidence that suggests discreteness of minimizers, leading us to the following
conjecture:

######  Conjecture 1.1.

Let $p>0$ and $p\not\in 2\mathbb{N}$. Then every minimizer of the $p$-frame
energy (1.4) is a finite discrete measure on $\mathbb{S}^{d-1}$.

There are other conjectures in the literature also asserting the discreteness
of measures minimizing certain energies on the sphere. We mention a couple of
examples.

Let $f(t)=\arccos|t|$, i.e. $f(\langle x,y\rangle)$ represents the non-obtuse
angle between the lines generated by the vectors $x$ and $y$. A conjecture of
Fejes Tóth [FT] states that the $N$-point energy (1.1) (the sum of acute
angles) is maximized by the periodically repeated elements of the orthonormal
basis, and the continuous version of the conjecture speculates that $I_{f}$ is
maximized by the discrete measure uniformly distributed over the elements of
the orthonormal basis (see [BM] for more details and recent results).

Another similar conjecture stems from mathematical physics and relativistic
quantum field theory [FS, BFSM]. It concerns the causal variational principle,
which, in the spherical case, concerns minimizing the energy on
$\mathbb{S}^{2}$ with the kernel

(1.5) $f(\langle x,y\rangle)=\max\\{0,\,2\tau^{2}(1+\langle
x,y\rangle)\big{(}2-\tau^{2}(1-\langle x,y\rangle)\big{)}\big{\\}},$

with a real parameter $\tau>0$. It is conjectured in [FS] that for any
$\tau\geq 1$ there exists a discrete minimizer, and for $\tau>\sqrt{2}$ all
minimizers are discrete. In [BGM+] it is demonstrated that for two values of
$\tau$, minimizers are the cross-polytope and the icosahedron, respectively.

In the present paper we prove a series of results which establish discreteness
of minimizers or smallness of their support (or at least the existence of such
minimizers) for various classes of energies on $\mathbb{S}^{d-1}$. In
particular, in Theorem 3.3 we prove a quantitative version of the following
statement:

######  Theorem 1.2.

Assume that $f\in C[-1,1]$ has only finitely many positive coefficients in its
orthogonal expansion with respect to Gegenbauer polynomials $C_{n}^{\lambda}$
(with $\lambda=\frac{d-2}{2}$). Then there exists a discrete minimizer of the
energy $I_{f}$ on $\mathbb{S}^{d-1}$.

The cardinality of the support of this discrete minimizer is bounded by the
dimension of the space of spherical harmonics, corresponding to the positive
coefficients of $f$. The proof relies on the analysis of the structure of
extreme points of the set of moment-constrained measures. Section 3 contains a
self-contained exposition of these arguments.

While the discreteness of the minimizers claimed in Conjecture 1.1 remains out
of reach, we establish that the support of the measures minimizing the
$p-$frame energy with $p\not\in 2\mathbb{N}$ must be small:

######  Theorem 1.3.

Assume that $p>0$ and $p\not\in 2\mathbb{N}$, and set $f(t)=|t|^{p}$. Let
$\mu\in\mathcal{P}(\mathbb{S}^{d-1})$ be a minimizer of the $p$-frame energy
$I_{f}$ (1.4). Then the support of $\mu$ has empty interior, i.e.

$\big{(}\operatorname{supp}\mu\big{)}^{\circ}=\emptyset.$

Section 4 is devoted to the proof of this theorem. In order to compare this
theorem to some known results on $\mathbb{R}^{d}$, we point out that
discreteness of minimizers for attractive-repulsive potentials on
$\mathbb{R}^{d}$ has been proved in [CFP] under the assumption that $f$ is
mildly repulsive, i.e. that the potential, as a function of $r=|x-y|$, behaves
as $-r^{\alpha}$ for small $r$, with $\alpha>2$. Since on the sphere
$\displaystyle{|\langle x,y\rangle|^{p}\approx 1-\frac{p}{2}r^{2}}$, the
$p$-frame potential corresponds to the endpoint case $\alpha=2$ and thus is
quite delicate: indeed, we know for some values of $p$ there exist non-
discrete minimizers. In the recent paper [LM] it was shown that for some
specific attractive-repulsive potentials with $\alpha\geq 2$, the
corresponding energies are uniquely minimized by discrete measures on regular
simplices. The complete understanding of the endpoint case $\alpha=2$ remains
an interesting open problem.

In Section 5, we also prove that an analog of Theorem 1.3 holds for energies
with kernels $f:[-1,1]\rightarrow\mathbb{R}$, which are real-analytic, but not
positive definite on $\mathbb{S}^{d-1}$ up to an additive constant (see
Definition 2.1 and Proposition 2.2). Theorem 5.1 states that for such kernels,
minimizing measures have support with empty interior. Moreover, on the circle
$\mathbb{S}^{1}$, they are discrete. This generates a certain dichotomy: for
an analytic functions $f$, either the energy $I_{f}$ is minimized by the
uniform surface measure $\sigma$, or all minimizers have support with empty
interior.

This result, as well as Theorem 1.2, obviously applies to polynomials. Thus,
when a polynomial $f$ is not positive definite (up to an additive constant),
the support of every minimizer has empty interior, while for every polynomial
$f$ there exists a discrete minimizer of $I_{f}$ (see Corollary 6.1). For
positive definite polynomials $f$, discrete minimizers are just weighted
spherical designs, but for arbitrary polynomials, existence of discrete
minimizers is new. Section 6 presents a discussion of energies with polynomial
kernels.

Finally, in Section 7 we present an interesting observation that for positive
definite kernels $f$, any local minimizer of the energy $I_{f}$ is necessarily
a global minimizer. This applies, in particular, to the $p$-frame energy with
even integer values of $p$ and to many other interesting energies.

##  2\. Background

###  2.1. Spherical harmonics and Gegenbauer polynomials

For a parameter $\lambda>0$, consider the weight
$\displaystyle{\nu(t)=(1-t^{2})^{\lambda-\frac{1}{2}}}$ on the interval
$[-1,1]$, where from now on $\lambda=\frac{d-2}{2}$. The weight
$\displaystyle{\nu(t)}$ is related to integration on the sphere
$\mathbb{S}^{d-1}$ in the following way: for a unit vector
$p\in\mathbb{S}^{d-1}$,

(2.1) $\int_{\mathbb{S}^{d-1}}f(\langle
x,p\rangle)d\sigma(x)=I_{f}(\sigma)=\frac{\omega_{d-2}}{\omega_{d-1}}\int_{-1}^{1}f(t)\big{(}1-t^{2}\big{)}^{\frac{d-3}{2}}\,dt,$

where, as before, $\sigma$ is the normalized surface measure on the sphere
$\mathbb{S}^{d-1}$ and $\omega_{d-1}=\frac{2\pi^{d/2}}{\Gamma(d/2)}$ is the
$(d-1)$-dimensional Hausdorff surface measure of $\mathbb{S}^{d-1}$.

Gegenbauer polynomials $C_{n}^{\lambda}$, $n\geq 0$, form a sequence of
orthogonal polynomials with respect to the weight $\nu(t)$ on the interval
$[-1,1]$. Every function $f\in L^{1}\big{(}[-1,1],\nu(t)dt\big{)}$ has a
Gegenbauer (ultraspherical) expansion

(2.2)
$f(t)\sim\sum_{n=0}^{\infty}\widehat{f}_{n}\frac{n+\lambda}{\lambda}C_{n}^{\lambda}(t).$

For $f\in L^{2}\big{(}[-1,1],\nu(t)dt\big{)}$ this expansion converges to $f$
in the $L^{2}$ sense. In the case of $\mathbb{S}^{1}$, when $\lambda=0$, the
relevant polynomials are the Chebyshev polynomials of the first kind

(2.3) $T_{n}(t)=\cos\big{(}n\arccos
t\big{)}=\frac{1}{2}\lim_{\lambda\rightarrow
0}\frac{n+\lambda}{\lambda}C_{n}^{\lambda}(t),$

and for $\mathbb{S}^{2}$, the polynomials are appropriately scaled Legendre
polynomials [Sz].

Let $\mathcal{H}_{n}^{d}$ denote the space of spherical harmonics of order
$n$, the functions which are restrictions to $\mathbb{S}^{d-1}$ of homogeneous
harmonic polynomials of degree $n$ on $\mathbb{R}^{d}$. These spaces are
mutually orthogonal for different values of $n$ and satisfy

$L^{2}\big{(}\mathbb{S}^{d-1},d\sigma\big{)}=\bigoplus_{n=0}^{\infty}\mathcal{H}_{n}^{d}.$

Let $\\{Y_{n,j}\\}$ be any orthonormal basis in $\mathcal{H}_{n}^{d}$. The
Gegenbauer polynomials are related to the spherical harmonics by the following
addition formula

(2.4)
$\sum_{j=1}^{a_{n}^{d}}Y_{n,j}(x)Y_{n,j}(y)=\frac{n+\lambda}{\lambda}C_{n}^{\lambda}(\langle
x,y\rangle)\ \ \textup{ for all }\,\,x,y\in\mathbb{S}^{d-1},$

where

$a_{n}^{d}=\text{dim}\
\mathcal{H}_{n}^{d}=\frac{n+\lambda}{\lambda}C_{n}^{\lambda}(1)=\frac{2n+d-2}{n+d-2}{{n+d-2}\choose{d-2}}.$

For more detailed information on spherical harmonics, Gegenbauer polynomials,
and harmonic analysis on the sphere, we refer the reader to [DX, Mu].

###  2.2. Positive definite functions on $\mathbb{S}^{d-1}$.

Positive definite functions play an important role in energy minimization.

######  Definition 2.1.

A function $f\in C[-1,1]$ is positive definite on subset $K$ of sphere
$\mathbb{S}^{d-1}$, if for every collection of points
$\\{x_{i}\\}_{i=1}^{N}\subset K$, the matrix $\big{[}f\big{(}\langle
z_{i},z_{j}\rangle\big{)}\big{]}_{i,j=1}^{N}$ is positive semidefinite, i.e.
for any sequence $\\{c_{i}\\}_{i=1}^{N}\subset\mathbb{C}$, $f$ satisfies the
inequality

$\sum\limits_{i,j=1}^{N}c_{i}\overline{c}_{j}f\big{(}\langle
x_{i},x_{j}\rangle\big{)}\geq 0.$

When $K=\mathbb{S}^{d-1}$ is the entire sphere, positive definite functions
admit several equivalent characterizations, which connect this property to
Gegenbauer polynomials and energy.

######  Proposition 2.2.

Let $f\in C[-1,1]$. The following statements are equivalent:

  1. (i)

The function $f$ is positive definite on $\mathbb{S}^{d-1}$.

  2. (ii)

For any signed Borel measure $\nu$ on $\mathbb{S}^{d-1}$, $I_{f}(\nu)\geq 0.$

  3. (iii)

The coefficients in the ultraspherical expansion (2.2) of $f$ with respect to
Gegenbauer polynomials $C_{n}^{\lambda}$ are non-negative:
$\widehat{f}_{n}\geq 0\,\,\textup{ for all }\,\,n\geq 0.$

  4. (iv)

The minimum of the energy $I_{f}$ over Borel probability measures on
$\mathbb{S}^{d-1}$ is achieved by $\sigma$ and is non-negative, i.e.

$\min_{\mu\in\mathcal{P}(\mathbb{S}^{d-1})}I_{f}(\mu)=I_{f}(\sigma)\geq 0.$

Part (iii) of this proposition is a classical result due to Schoenberg [Sch].
Part (iv) states that positive definite functions (up to additive constants)
are precisely those potentials for which energy minimization imposes uniform
distribution.

Moreover, positive definiteness implies uniform convergence of the Gegenbauer
expansion (2.2); see e.g. [G].

######  Lemma 2.3.

Assume that $f\in C[-1,1]$ is positive definite on $\mathbb{S}^{d-1}$. Then
the Gegenbauer expansion (2.2) converges to $f$ absolutely and uniformly on
$[-1,1]$.

One of the simplest ways to prove this statement is using Mercer’s theorem
from spectral theory on the representation of symmetric positive definite
functions [Me]. In turn, Lemma 2.3 together with the addition formula (2.4)
easily imply part (iv) of Proposition 2.2.

Positive definiteness also plays a role when the energy is not minimized by
the uniform measure $\sigma$. In this case, we have the following implication
[Bj, FS].

######  Lemma 2.4.

Let $f\in C([-1,1])$. Assume that $\mu$ is a minimizer of $I_{f}$ over
$\mathcal{P}(\mathbb{S}^{d-1})$ and $I_{f}(\mu)\geq 0$. Then the function $f$
must be positive definite on $\operatorname{supp}(\mu)$.

Observe that, together with part (iv) of Proposition 2.2, this immediately
implies the following:

######  Corollary 2.5.

Either $\sigma$ is a minimizer of $I_{f}$ (i.e. $f$ is positive definite on
$\mathbb{S}^{d-1}$, up to an additive constant), or every minimizer of $I_{f}$
is supported on a proper subset of the sphere $\mathbb{S}^{d-1}$.

Much of this paper is dedicated to obtaining various refinements of this
principle for various classes of kernels $f$. Lemma 2.4 also suggests an
approach to proving that a certain set cannot be contained in the support of a
minimizer: one may attempt to prove that $f$ is not positive definite on that
set. This idea, albeit not in a straightforward fashion, is exploited in the
proof of Theorem 1.3 in the next section; see the proof of Proposition 4.2.

###  2.3. Gegenbauer expansions and other minimizers

In some situations, Gegenbauer coefficients can give some information about
the minimizers, even when $\sigma$ does not minimize the energy. Below we
mention several relevant results of this type. While we do not use them in
this paper, we chose to include them because they are similar in spirit to the
results of the paper: they provide certain conditions, under which there exist
discrete minimizers or all minimizers are discrete. These results can be found
in [BiD].

  * •

If $\widehat{f}_{n}\leq 0$ for all $n\geq 1$, then a Dirac delta mass
$\mu=\delta_{z}$, for any $z\in\mathbb{S}^{d-1}$, is a minimizer of $I_{f}$.
If $f$ has a strict absolute minimum at $t=1$ (in particular, if
$\widehat{f}_{n}<0$ for all $n\geq 1$), then every minimizer is a Dirac mass.
Observe that this case resonates with Theorem 3.3.

  * •

If $(-1)^{n+1}\widehat{f}_{n}\geq 0$ for all $n\geq 1$, then the measure
$\mu=\frac{1}{2}\big{(}\delta_{z}+\delta_{-z}\big{)}$ is a minimizer of
$I_{f}$. Moreover, all minimizers are of this form, if the strict inequality
$(-1)^{n+1}\widehat{f}_{n}>0$ holds.

  * •

If $\widehat{f}_{2n}=0$ and $\widehat{f}_{2n-1}\geq 0$ for all $n\geq 1$, then
every centrally symmetric measure minimizes $I_{f}$. In particular, there
exist discrete minimizers.

We note that for the Euclidean setting, and certain attractive-repulsive
potentials, there are classifications of potentials for which two-point
measures appear as minimizers, see [KKLS].

##  3\. Existence of discrete minimizers

###  3.1. Extreme points for sets of moment-constrained measures

In the present section we exhibit a large class of potentials $f$ for which
there exist discrete minimizers of the energies $I_{f}$. The methods that we
employ are closely related to moment problems.

Let $\Omega$ be a compact metric space and let $\mathcal{B}_{+}(\Omega)$
denote the set of positive Borel measures on $\Omega$. Given continuous
functions $f_{0},...,f_{n}$ on $\Omega$ and non-negative constants $c_{i}$, we
consider the set

(3.1)
$K=\bigg{\\{}\mu\in\mathcal{B}_{+}(\Omega):\int_{\Omega}f_{i}d\mu=c_{i},\,i=0,1,\dots,n\bigg{\\}},$

which consists of Borel measures whose moments with respect to $f_{i}\in
C(\Omega)$ are fixed. We always set $f_{0}\equiv 1$ and $c_{0}=1$, so that
$\mu\in K$ guarantees that $\mu$ is a probability measure, i.e.
$\mu(\Omega)=1$.

It is easy to see that $K$ is convex, bounded, and weak-$*$ closed, and
therefore is weak-$*$ compact. By the Krein–Milman theorem, $K$ is the
weak-$*$ closure of $\operatorname{ext}(K)$ — the set of extreme points of
$K$. The results presented below describe the structure of
$\operatorname{ext}(K)$, in particular, the discreteness of its elements. To
make this section self-contained, we include their proofs.

We start with a theorem which gives a necessary condition for $\mu$ to be an
extreme point of $K$.

######  Theorem 3.1 (Douglas, [D]).

Assume that $\mu\in\operatorname{ext}(K)$. Then

(3.2) $L^{1}(d\mu)=\operatorname{span}\\{f_{0}=1,f_{1},\dots,f_{n}\\}.$

###### Proof.

Assume that $g\in L^{\infty}(d\mu)$ satisfies

$\int_{\Omega}f_{i}gd\mu=0,\qquad i=0,1,\dots,n.$

Multiplying $g$ by a constant, we may assume that
$\|g\|_{L^{\infty}(d\mu)}<1$. Then the measures $\mu_{\pm}$, defined by
$d\mu_{\pm}=(1\pm g)d\mu$, belong to $K$, since
$\mu_{\pm}\in\mathcal{B}_{+}(\Omega)$ and

$\int_{\Omega}f_{i}d\mu_{\pm}=\int_{\Omega}f_{i}(1\pm
g)d\mu=\int_{\Omega}f_{i}d\mu=c_{i}.$

At the same time, $\mu=\frac{1}{2}(\mu_{-}+\mu_{+})$. Since
$\mu\in\operatorname{ext}K$, this implies that $\mu_{\pm}=\mu$ and hence $g=0$
$\mu$-a.e. Therefore, the functions $f_{i}$ span $L^{1}(d\mu)$. ∎

We now state and prove a result, which demonstrates the discreteness of the
elements of $\operatorname{ext}(K)$. This result has a number of precursors
and extensions, see [Ri, Ro1, Ro2, Ros, W, Zu].

######  Theorem 3.2 (Karr, [K]).

Let $\mu\in K$. Then the following statements are equivalent:

  1. (i)

$\mu\in\operatorname{ext}(K)$.

  2. (ii)

The cardinality of $\operatorname{supp}\mu$ is at most $n+1$. Moreover, if we
denote $\operatorname{supp}\mu=\\{x_{1},\dots,x_{k}\\}$, then the vectors
$v_{j}=\big{(}1,f_{1}(x_{j}),\dots,f_{n}(x_{j})\big{)}$, $j=1,2,\dots,k$, are
linearly independent.

###### Proof.

(i)$\Rightarrow$(ii). Assume that there exist points
$\\{x_{1},\dots,x_{n+2}\\}\subset\operatorname{supp}\mu$. Then one can find a
vector $y\in\mathbb{R}^{n+2}$, which is not in the span of the vectors
$\big{(}f_{i}(x_{1}),f_{i}(x_{2}),\dots,f_{i}(x_{n+2})\big{)}$,
$i=0,1,\dots,n$, since the latter subspace is at most $n+1$ dimensional.
Appealing to Urysohn’s lemma, one can construct a function $g\in
C(\Omega)\subset L^{1}(d\mu)$ such that $g(x_{i})=y_{i}$ for
$i=1,2,\dots,n+2$. But then $g\not\in\operatorname{span}\\{f_{i}\\}$, which
contradicts Theorem 3.1, i.e. $\big{|}\operatorname{supp}\mu\big{|}\leq n+1$.

Now that it is known that $\mu=\sum_{i=1}^{k}t_{i}\delta_{x_{i}}$ with $k\leq
n+1$, $t_{i}>0$, $\sum t_{i}=1$, consider the linear system

(3.3) $\begin{pmatrix}1&\dots&1\\\ f_{1}(x_{1})&\dots&f_{1}(x_{k})\\\
\vdots&\ddots&\vdots\\\
f_{n}(x_{1})&\dots&f_{n}(x_{k})\end{pmatrix}\begin{pmatrix}\alpha_{1}\\\
\alpha_{2}\\\ \vdots\\\ \alpha_{k}\end{pmatrix}=\begin{pmatrix}1\\\ c_{1}\\\
\vdots\\\ c_{n}\end{pmatrix}.$

This system has a unique solution $\alpha_{i}=t_{i}$, since if the solution is
not unique, then there is a whole affine subspace of solutions and one could
perturb the values of $t_{i}$ in opposite directions, i.e. find two solutions
of the form $\\{t_{i}\pm\tau_{i}\\}$, and construct two measures
$\mu_{\pm}=\sum_{i=1}^{k}(t_{i}\pm\tau_{i})\delta_{x_{i}}$ so that
$\mu_{\pm}\geq 0$ and $\int f_{i}d\mu_{\pm}=\int f_{i}d\mu$, i.e.
$\mu_{\pm}\in K$, and $\mu=\frac{1}{2}(\mu_{+}+\mu_{-})$, which contradicts
the fact that $\mu\in\operatorname{ext}(K)$. This proves the linear
independence of the rows of the matrix above.

(ii)$\Rightarrow$(i). Assume that (ii) holds. Then the system (3.3) has a
unique solution, i.e. $\mu$ is uniquely determined by the the condition
$\operatorname{supp}\mu\subset\\{x_{1},\dots,x_{k}\\}$. If
$\mu=\frac{1}{2}(\mu_{1}+\mu_{2})$, then
$\operatorname{supp}\mu\subset\operatorname{supp}\mu_{1}\cup\operatorname{supp}\mu_{2}$,
and thus $\operatorname{supp}\mu_{j}\subset\\{x_{1},\dots,x_{k}\\}$ for
$j=1,2$. Therefore $\mu_{1}=\mu_{2}=\mu$, i.e. $\mu\in\operatorname{ext}(K)$.
∎

We remark that convex geometry plays heavily into similar characterizations of
solutions to infinite dimensional optimization problems in the recent papers
[BCD+, CRPW, UFW].

###  3.2. Applications of Karr’s theorem: existence of discrete minimizers.

We now apply the results on moment-constrained measures to prove that for a
function $f$ with only finitely many positive terms in its Gegenbauer
expansion, there exist discrete minimizers of $I_{f}$.

Let $\widehat{f}_{n}$ denote the coefficients in the Gegenbauer expansion
(2.2) of the function $f\in C[-1,1]$. Consider the sets $N_{+}(f)=\\{n\geq
0:\widehat{f}_{n}>0\\}$ and $N_{-}(f)=\\{n\geq 0:\widehat{f}_{n}<0\\}$. We
shall assume that

(3.4) $\big{|}N_{+}(f)\big{|}<\infty,$

i.e. there are only finitely many terms of (2.2) with $\widehat{f}_{n}>0$. In
this case, the function

$\displaystyle{\sum_{n\in
N_{+}(f)}\widehat{f}_{n}\,\frac{n+\lambda}{\lambda}C_{n}^{\lambda}(t)-f(t)}$

is continuous and positive definite. According to Lemma 2.3, this implies that
the Gegenbauer expansion (2.2) of $f$ converges uniformly and absolutely.

Recall that $\mathcal{H}_{n}^{d}$ denotes the space of spherical harmonics of
degree $n$ on $\mathbb{S}^{d-1}$. We are now ready to state the main theorem
of the section.

######  Theorem 3.3.

Assume that the Gegenbauer expansion (2.2) of the function $f\in C[-1,1]$
satisfies

$\big{|}N_{+}(f)\big{|}=\big{|}\\{n\geq 0:\widehat{f}_{n}>0\\}\big{|}<\infty,$

i.e. the Gegenbauer expansion has only finitely many positive terms. Then
there exists a discrete measure $\mu^{*}\in\mathcal{P}(\mathbb{S}^{d-1})$ such
that

(3.5) $\big{|}\operatorname{supp}\mu^{*}\big{|}\leq\sum_{n\in
N_{+}(f)\cup\\{0\\}}\operatorname{dim}\mathcal{H}_{n}^{d},$

and $\mu^{*}$ minimizes the energy $I_{f}(\mu)$ over
$\mathcal{P}(\mathbb{S}^{d-1})$, i.e.

(3.6) $I_{f}(\mu^{*})=\inf_{\mu\in\mathcal{P}(\mathbb{S}^{d-1})}I_{f}(\mu).$

###### Proof.

Let $\nu\in\mathcal{P}(\mathbb{S}^{d-1})$ be any minimizer of $I_{f}$ and set

$\displaystyle{M=\inf_{\mu\in\mathcal{P}(\mathbb{S}^{d-1})}I_{f}(\mu)=I_{f}(\nu)}.$

We shall use the addition formula for spherical harmonics (2.4), as well as
the absolute convergence of (2.2), to re-write for any measure
$\mu\in\mathcal{P}(\mathbb{S}^{d-1})$, $I_{f}(\mu)$ as,

$\displaystyle\sum_{n=0}^{\infty}\widehat{f}_{n}\int_{\mathbb{S}^{d-1}}\int_{\mathbb{S}^{d-1}}\frac{n+\lambda}{\lambda}\
C_{n}^{\lambda}(\langle
x,y\rangle)d\mu(x)d\mu(y)=\sum_{n=0}^{\infty}\widehat{f}_{n}\Bigg{[}\sum_{j=1}^{\operatorname{dim}\mathcal{H}_{n}^{d}}\bigg{(}\int_{\mathbb{S}^{d-1}}Y_{n,j}(x)d\mu(x)\bigg{)}^{2}\Bigg{]}$
$\displaystyle=\sum_{n\in
N_{+}(f)}\widehat{f}_{n}\Bigg{[}\sum_{j=1}^{\operatorname{dim}\mathcal{H}_{n}^{d}}\bigg{(}\int_{\mathbb{S}^{d-1}}Y_{n,j}(x)d\mu(x)\bigg{)}^{2}\Bigg{]}-\sum_{n\in
N_{-}(f)}\big{(}-\widehat{f}_{n}\big{)}\Bigg{[}\sum_{j=1}^{\operatorname{dim}\mathcal{H}_{n}^{d}}\bigg{(}\int_{\mathbb{S}^{d-1}}Y_{n,j}(x)d\mu(x)\bigg{)}^{2}\Bigg{]},$

the last of which we define as the difference of functionals
$\mathcal{F}(\mu)-\mathcal{G}(\mu)$. It is easy to see that $\mathcal{G}$ is
convex with respect to $\mu$ since it is a positive linear combination of
squares of linear functionals of $\mu$. Let us set

$K=\Bigg{\\{}\mu\in\mathcal{B}_{+}(\mathbb{S}^{d-1}):\int_{\mathbb{S}^{d-1}}Y_{n,j}d\mu(x)=\int_{\mathbb{S}^{d-1}}Y_{n,j}d\nu(x),\,n\in
N_{+}(f),\,j=1,2,\dots,\operatorname{dim}\mathcal{H}_{n}^{d}\Bigg{\\}},$

so that $\nu\in K$ and $\mathcal{F}(\mu)=\mathcal{F}(\nu)$ for $\mu\in K$.
Without loss of generality, we shall assume that $0\in N_{+}(f)$. This
guarantees that $\mu\in K$ is a probability measure (similarly to setting
$c_{0}=1$ and $f_{0}\equiv 1$ earlier). Since $N_{+}(f)<\infty$, the set $K$
has finitely many moment constraints and Theorem 3.2 is applicable. In fact,
the number of constraints is exactly the right-hand side of (3.5).

Given that $\mathcal{G}$ is convex in $\mu$ and $K$ is a convex weak-$*$
compact subset of $\mathcal{B}_{+}(\mathbb{S}^{d-1})$, we conclude that
$\mathcal{G}(\mu)$ achieves its maximum on $K$ at a point of
$\operatorname{ext}(K)$. Hence there exists a measure
$\mu^{*}\in\operatorname{ext}(K)$ such that
$\displaystyle{\mathcal{G}(\mu^{*})=\sup_{\mu\in K}\mathcal{G}(\mu)}$. We then
find that

$\displaystyle M$
$\displaystyle=I_{f}(\nu)=\mathcal{F}(\nu)-\mathcal{G}(\nu)=\mathcal{F}(\mu^{*})-\mathcal{G}(\nu)\geq\mathcal{F}(\mu^{*})-\mathcal{G}(\mu^{*})=I_{f}(\mu^{*})\geq
M,$

i.e. $I_{f}(\mu^{*})=M$ and $\mu^{*}$ is also a minimizer of $I_{f}$.

Since $\mu^{*}\in\operatorname{ext}(K)$, we can apply Karr’s theorem (Theorem
3.2) to finish the proof of the theorem. ∎

##  4\. Empty interior of $p$-frame energy minimizers: the proof of Theorem
1.3

Conjecture 1.1, stating that the minimizers of the $p$-frame energy with
$p\not\in 2\mathbb{N}$ are necessarily discrete, remains open, outside of some
specific cases covered in our companion paper [BGM+]. In the present section,
we prove a weaker statement, namely, that the support of every minimizer of
such energies has empty interior, i.e. Theorem 1.3.

A similar result has been proved in [FS] for the energy on $\mathbb{S}^{2}$
with the kernel given by (1.5). While our approach is inspired by theirs and
the main line of reasoning follows an analogous path, specific constructions
and arguments in the proofs of Propositions 4.2 and 4.3 below are much more
peculiar and significantly more involved in the case of the $p$-frame energy.

We shall need a standard fact from potential theory ([Bj, BHS, L]). For a
measure $\mu\in\mathcal{P}(\mathbb{S}^{d-1})$, let us define the potential
$F_{\mu}$ of $\mu$ with respect to $f$ as

(4.1) $F_{\mu}(x):=\int_{\mathbb{S}^{d-1}}f(\langle
x,y\rangle)d\mu(y),\,\,\,x\in\mathbb{S}^{d-1}.$

Notice that this meaning of the term “potential” is consistent with our
previous usage, since the function $f(\langle x,y\rangle)$ is just the
potential generated by a unit point charge at $y$, i.e. $f(\langle
x,y\rangle)=F_{\delta_{y}}(x)$. It is a well-known phenomenon from
electrostatics that the potential of the equilibrium measure is constant on
the support of the measure.

######  Lemma 4.1.

If $f\in C([-1,1])$ and $\mu$ is a minimizer of $I_{f}$, then the potential
$F_{\mu}$ is constant on the support of $\mu$:

(4.2)
$F_{\mu}\big{|}_{\operatorname{supp}\mu}=\inf_{x\in\mathbb{S}^{d-1}}F_{\mu}(x)=I_{f}(\mu).$

In what follows, the value of $p\in\mathbb{R}_{+}\setminus 2\mathbb{N}$ is
fixed, $f(t)=|t|^{p}$, and $\mu$ is assumed to be a minimizer of $I_{f}$. The
proof of Theorem 1.3 is based on two properties of interior points of
$\operatorname{supp}(\mu)$.

######  Proposition 4.2.

Let $p\in\mathbb{R}_{+}\setminus 2\mathbb{N}$, $f(t)=|t|^{p}$, and $\mu$ be a
minimizer of $I_{f}$. Then for
$z\in\big{(}\operatorname{supp}\mu\big{)}^{\circ}$,

$\operatorname{supp}\mu\cap z^{\perp}=\emptyset.$

######  Proposition 4.3.

Let the same conditions as in Proposition 4.2 hold. Then for
$z\in\big{(}\operatorname{supp}\mu\big{)}^{\circ}$,

$\operatorname{supp}\mu\cap z^{\perp}\neq\emptyset.$

Since these two statements are clearly mutually exclusive whenever
$\operatorname{supp}\mu$ is non-empty, their validity proves Theorem 1.3, i.e.
that there are no interior points in the support of a minimizer. The remainder
of this section is dedicated to the proof of these propositions.

We now sketch the argument for the first proposition. In short, the idea of
the proof is the following. Assume that there exists a point
$y\in\operatorname{supp}\mu$ such that $\langle y,z\rangle=0$. We shall
construct a finite set of points
$X=\big{\\{}x_{i}\big{\\}}_{i=1}^{N}\subset\operatorname{supp}\mu$, such that
the matrix $\big{[}f(\langle x_{i},x_{j}\rangle)\big{]}_{i,j}$ is not positive
semidefinite, thus violating Lemma 2.4. The set $X$ will consist of the points
$z$, $y$, and a number (depending on $p$) of points, equidistantly spaced
around $z$ on the great circle connecting $y$ and $z$. We now make this
precise.

###### The proof of Proposition 4.2.

We prove that if $z$ is an interior point of a minimizer’s support, then the
orthogonal hyperplane $z^{\perp}$ does not intersect the support of $\mu$.

Fix $z$ in the interior of $\operatorname{supp}(\mu)$ and let
$y\in\mathbb{S}^{d-1}$ be any point such that $\langle y,z\rangle=0$. Setting
$k\in\mathbb{N}$ so that $2k-2<p<2k$, we shall construct a set
$\\{x_{0},\dots,x_{N-1}\\}$ of $N=2k+2$ points, all of which lie on the great
circle connecting $z$ and $y$. The points $x_{0},\dots,x_{2k}$ are chosen in
such a way that the angle between $x_{j}$ and $z$ is $(j-k)\varepsilon$ for
some small $\varepsilon>0$. Thus $x_{k}=z$, and the points $x_{0}$ and
$x_{2k}$ make angles $-k\varepsilon$ and $k\varepsilon$ with $z$,
respectively. Observe that when $\varepsilon$ is small enough, all of these
points $x_{0},\dots,x_{2k}$ belong to $\operatorname{supp}(\mu)$, since $z$ is
an interior point. Finally, we set $x_{2k+1}=y$. Then the angle between
$x_{2k+1}=y$ and $x_{j}$, $j=0,\dots,2k$, is $\frac{\pi}{2}-(j-k)\varepsilon$.
In order to apply Lemma 2.4, we consider the matrix
$\displaystyle{A=\big{[}f(\langle x_{i},x_{j}\rangle)\big{]}_{i,j=0}^{2k+1}}$.

We will show that the matrix $A$ is not positive semidefinite. To this end, we
first construct an auxiliary vector $v\in\mathbb{R}^{2k+1}\setminus\\{0\\}$
such that for $m\in\\{0,1,...,2k-1\\}$,

(4.3) $\sum_{j=0}^{2k}j^{m}v_{j}=0,$

i.e. this vector must be in the (right) kernel of the Vandermonde matrix

$\begin{pmatrix}1&1&1&1&\cdots&1\\\ 0&1&2&3&\cdots&2k\\\
0&1&2^{2}&3^{2}&\cdots&(2k)^{2}\\\
\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\\
0&1&2^{2k-1}&3^{2k-1}&\cdots&(2k)^{2k-1}\end{pmatrix}.$

We can take the entries of $v$ to be

(4.4) $v_{j}=\prod_{\begin{subarray}{c}i=0\\\ i\neq
j\end{subarray}}^{2k}\frac{1}{j-i}=\frac{(-1)^{j}}{(2k-j)!j!}.$

Such a vector can be seen to be in the kernel of the matrix above by use of
the formula for the inverse of the square Vandermonde matrix (see Ex. 40 on
page 38 of [Kn]).

Consider a vector $u=[\alpha v_{0},\alpha v_{1},...,\alpha
v_{2k},\beta]^{T}\in\mathbb{R}^{2k+2}$, where $\alpha$, $\beta\in\mathbb{R}$.
Then we have

(4.5) $\displaystyle\left\langle Au,u\right\rangle$
$\displaystyle=\alpha^{2}\left(\sum_{i,j=0}^{2k}v_{i}v_{j}f(\langle
x_{i},x_{j}\rangle)\right)+2\alpha\beta\left(\sum_{j=0}^{2k}v_{j}f(\langle
x_{2k+1},x_{j}\rangle)\right)+\beta^{2}.$

We shall show that the real numbers $\alpha$ and $\beta$ can be chosen in such
a way that the expression above is negative, for $\varepsilon$ sufficiently
small.

Observe that for $i$, $j=0,\dots,2k$ we have

$f(\langle x_{i},x_{j}\rangle)=\cos^{p}\big{(}(i-j)\varepsilon\big{)}.$

Since $\cos^{p}(t)$ is even, smooth near zero, and $\cos^{p}(0)=1$, we can use
its Taylor expansion to estimate the first term of (4.5) as follows

(4.6) $\displaystyle\sum_{i,j=0}^{2k}v_{i}v_{j}f(\langle x_{i},x_{j}\rangle)$
$\displaystyle=\sum_{i,j=0}^{2k}v_{i}v_{j}\cos^{p}\big{(}(i-j)\varepsilon\big{)}$
$\displaystyle=\sum_{i,j=0}^{2k}v_{i}v_{j}\left(1+\sum_{m=1}^{2k-1}a_{m}\varepsilon^{2m}(i-j)^{2m}+O(\varepsilon^{4k})\right)$
$\displaystyle=\left(\sum_{j=0}^{2k}v_{j}\right)\left(\sum_{i=0}^{2k}v_{i}\right)+\sum_{m=1}^{2k-1}a_{m}\varepsilon^{2m}\left(\sum_{i,j=0}^{2k}v_{i}v_{j}(i-j)^{2m}\right)+O(\varepsilon^{4k})$
$\displaystyle=\sum_{m=1}^{2k-1}a_{m}\varepsilon^{2m}\sum_{i,j=0}^{2k}v_{i}v_{j}\sum_{l=0}^{2m}\binom{2m}{l}i^{l}j^{2m-l}+O(\varepsilon^{4k})$
$\displaystyle=\sum_{m=1}^{2k-1}a_{m}\varepsilon^{2m}\sum_{l=0}^{2m}\binom{2m}{l}\left(\sum_{i=0}^{2k}v_{i}i^{l}\right)\left(\sum_{j=0}^{2k}v_{j}j^{2m-l}\right)+O(\varepsilon^{4k})$
$\displaystyle=O(\varepsilon^{4k}),$

where we have used the fact that for all values of $l=0,1,\dots,2m$, either
$l\leq 2k-1$ or $2m-l\leq 2k-1$.

We now turn to the second term of (4.5). Observe that for $j=0,\dots,2k$ we
have

(4.7) $f(\langle x_{2k+1},x_{j}\rangle)=f(\langle
y,x_{j}\rangle)=\cos^{p}\bigg{(}\frac{\pi}{2}-(j-k)\varepsilon\bigg{)}=\big{|}\sin\big{(}(j-k)\varepsilon\big{)}\big{|}^{p}.$

We then find that

(4.8) $\displaystyle\sum_{j=0}^{2k}v_{j}f(\langle y,x_{j}\rangle)$
$\displaystyle=\sum_{j=0}^{2k}v_{j}\sin^{p}\left(|k-j|\varepsilon\right)=\sum_{j=0}^{2k}v_{j}\left({|k-j|\varepsilon}+O(\varepsilon^{3})\right)^{p}$
$\displaystyle=\sum_{j=0}^{2k}v_{j}\left(|k-j|\varepsilon\right)^{p}\left(1+O(\varepsilon^{2})\right)^{p}=\varepsilon^{p}\sum_{j=0}^{2k}v_{j}|k-j|^{p}+O(\varepsilon^{p+2}).$

We now analyze the coefficient of $\varepsilon^{p}$ in the above expression
using (4.4)

$\displaystyle\sum_{j=0}^{2k}v_{j}|k-j|^{p}$
$\displaystyle=\sum_{j=0}^{2k}(-1)^{j}\frac{|k-j|^{p}}{(2k-j)!j!}=2\sum_{j=0}^{k-1}(-1)^{j}\frac{(k-j)^{p}}{(2k-j)!j!}.$

Since the above is a sum of $k$ exponential functions of $p$, we know that
$\sum_{j=0}^{k-1}(-1)^{j}\frac{(k-j)^{p}}{(2k-j)!j!}$ has at most $k-1$ zeros,
see e.g. Ex. 75 from [PS, pg. 46]. We will show that these zeros are exhausted
by the even integer values $p=2,4,\dots,2k-2$. Indeed, assume indirectly that

$2\sum_{j=0}^{k-1}(-1)^{j}\frac{(k-j)^{p}}{(2k-j)!j!}=b\neq 0$

for some even integer $0<p\leq 2k-2$. Then according to (4.5), (4.6), and
(4.8) we have

$\displaystyle\left\langle Au,u\right\rangle$
$\displaystyle=\alpha^{2}O(\varepsilon^{4k})+2\alpha\beta\left(b\varepsilon^{p}+O(\varepsilon^{p+2})\right)+\beta^{2}.$

Since $p<2k$, for $\varepsilon$ sufficiently small, the discriminant of this
quadratic form is positive, hence we can choose $\alpha$ and $\beta$ so that
$\langle Au,u\rangle<0$. However, since $f(t)=|t|^{p}$ is a positive definite
function on $\mathbb{S}^{d-1}$ for even integer $p$, this is a contradiction,
as the matrix $A$ must be positive semidefinite for any collection
$\\{x_{i}\\}$. Therefore

$2\sum_{j=0}^{k-1}(-1)^{j}\frac{(k-j)^{p}}{(2k-j)!j!}=0$

for all $p\in\\{2,4,...,2k-2\\}$. Since there are at most $k-1$ zeros of this
function, we then know that

$b_{p}:=2\sum_{j=0}^{k-1}(-1)^{j}\frac{(k-j)^{p}}{(2k-j)!j!}\neq 0$

for all other values of $p$. Let $p\in(0,2k)\setminus\\{2,4,...,2k-2\\}$. Then

$\displaystyle\left\langle Au,u\right\rangle$
$\displaystyle=\alpha^{2}O(\varepsilon^{4k})+2\alpha\beta\left(b_{p}\varepsilon^{p}+O(\varepsilon^{p+2})\right)+\beta^{2},$

and by the previous argument, for $\varepsilon$ sufficiently small, we could
choose $\alpha$ and $\beta$ so that $\langle Au,u\rangle<0$, i.e. $A$ is not
positive definite. Thus, according to Lemma 2.4,
$\\{x_{0},x_{1},...,x_{2k},y\\}$ is not a subset of $\operatorname{supp}\mu$.
Since, by assumption, for small $\varepsilon>0$ the points
$x_{0},x_{1},\dots,x_{2k}$ all lie in a neighborhood of $z$ and hence in
$\operatorname{supp}\mu$, this implies that $y\not\in\operatorname{supp}\mu$
and so $\operatorname{supp}\mu\cap z^{\perp}=\emptyset$. ∎

We would like to make the following remark. Observe that for $p\not\in
2\mathbb{N}$, the number of points used to disprove positive definiteness of
$f(t)=|t|^{p}$ in the argument above is of the order $p$. A restriction of
this type is actually necessary. Indeed, according to the result of Fitzgerald
and Horn [FH], for any positive definite matrix
$A=\big{[}a_{ij}\big{]}_{i,j=1}^{N}$ with non-negative entries $a_{ij}\geq 0$,
its Hadamard powers $A^{(\alpha)}=\big{[}a_{ij}^{\alpha}\big{]}_{i,j=1}^{N}$
are also positive definite when $\alpha\geq N-2$. Let $G=\big{[}\langle
x_{i},x_{j}\rangle\big{]}_{i,j=1}^{N}$ be the Gram matrix of the set
$X=\\{x_{1},\dots,x_{N}\\}\subset\mathbb{S}^{d-1}$. Since the matrix
$G^{(2)}=\big{[}|\langle x_{i},x_{j}\rangle|^{2}\big{]}_{i,j=1}^{N}$ is
positive definite and has non-negative entries, we have that the matrix

$G^{(p)}=\big{[}|\langle
x_{i},x_{j}\rangle|^{p}\big{]}_{i,j=1}^{N}=\big{(}G^{(2)}\big{)}^{(p/2)}$

is positive definite whenever $p/2\geq N-2$. Therefore, to obtain a non-
positive definite matrix $G^{(p)}$, we must take $N\geq 2+p/2$ points.

We now complete the proof of Theorem 1.3.

###### The proof of Proposition 4.3.

Suppose a neighborhood of a point $z\in\mathbb{S}^{d-1}$ is contained in the
support of $\mu$. We shall demonstrate that $\operatorname{supp}\mu$ must
intersect the hyperplane $z^{\perp}$.

Let us assume the contrary, i.e. $\operatorname{supp}\mu\cap
z^{\perp}=\emptyset$. We may move all the mass of $\mu$ to the hemisphere
centered at $z$ by defining a new measure
$\mu_{z}\in\mathcal{P}(\mathbb{S}^{d-1})$:

$\mu_{z}(E)=\begin{cases}\mu(-E\cup E),&\textup{if
}E\subseteq\\{x\in\mathbb{S}^{d-1}:\,\langle z,x\rangle>0\\},\\\
\mu(E),&\textup{if }E\subseteq z^{\perp},\\\ 0,&\textup{if
}E\subseteq\\{x\in\mathbb{S}^{d-1}:\,\langle z,x\rangle<0\\}.\end{cases}$

Since $f(\langle z,y\rangle)=f(\langle z,-y\rangle)$ for all
$y\in\mathbb{S}^{d-1}$, this does not change the energy, i.e.
$I_{f}(\mu_{z})=I_{f}(\mu)$, so that $\mu_{z}$ is also a minimizer.

Since $\displaystyle{\operatorname{supp}\mu\cap z^{\perp}=\emptyset}$, we also
have that $\displaystyle{\operatorname{supp}\mu_{z}\cap z^{\perp}=\emptyset}$,
i.e. $\operatorname{supp}\mu_{z}\subset\\{x\in\mathbb{S}^{d-1}:\,\langle
z,x\rangle>0\\}$. Compactness of the support of $\mu_{z}$ then implies that it
is separated from $z^{\perp}$, i.e. for some $\delta>0$ we have $\langle
y,z\rangle>\delta$ for each $y\in\operatorname{supp}\mu_{z}$. Let us choose an
open neighborhood $U_{z}$ of $z$, small enough so that
$U_{z}\subset\operatorname{supp}\mu_{z}$ and so that for each $x\in U_{z}$ and
each $y\in\operatorname{supp}\mu_{z}$, $\langle y,x\rangle>\delta>0$.

We can now write the potential (4.1) of $\mu_{z}$ at the point $x\in U_{z}$ as

(4.9) $F_{\mu_{z}}(x)=\int_{\mathbb{S}^{d-1}}|\langle
x,y\rangle|^{p}\,d\mu_{z}(y)=\int_{\operatorname{supp}\mu_{z}}\langle
x,y\rangle^{p}\,d\mu_{z}(y).$

The discussion above implies that the last expression is well-defined for all
$p>0$. According to Lemma 4.1, the potential $F_{\mu_{z}}(x)$ is constant on
$U_{z}\subset\operatorname{supp}\mu_{z}$.

When $p$ is an odd integer, the proof can be finished very quickly. Indeed, in
this case the expression

$g(x)=\int_{\operatorname{supp}\mu_{z}}\langle x,y\rangle^{p}\,d\mu_{z}(y)$

is well defined for each $x\in\mathbb{S}^{d-1}$ and yields an analytic
function on the sphere (actually, a polynomial). Hence, being constant on an
open set, implies that it is is constant on all of $\mathbb{S}^{d-1}$, which
is not possible since, obviously,
$g(-z)=-g(z)=-F_{\mu_{z}}(z)=-I_{f}(\mu_{z})\neq 0$. Compare this argument to
Theorem 5.1.

We now will present an approach which works for all
$p\in\mathbb{R}_{+}\setminus 2\mathbb{N}$. Assume that there exists a
differential operator $D$ acting on functions on the sphere with the following
two properties:

  1. (i)

$D$ locally annihilates constants, i.e. if $u(x)$ is constant on some open set
$\Omega$, then $D_{x}u=0$ on $\Omega$;

  2. (ii)

$D_{x}\left(\langle x,y\rangle^{p}\right)<0$ for all $x\in U_{z}$ and
$y\in\operatorname{supp}\mu_{z}$.

Existence of such an operator would finish the proof since we would then have
for each $x\in U_{z}$

(4.10)
$0=D_{x}F_{\mu_{z}}(x)=\int_{\operatorname{supp}(\mu_{z})}D_{x}\left(\langle
x,y\rangle^{p}\right)\,d\mu_{z}(y)<0,$

which is a contradiction. Note that that switching to $D_{x}\left(\langle
x,y\rangle^{p}\right)>0$ in condition (ii) does not affect the proof.

We now construct such an operator $D$. Let $\Delta$ denote the
Laplace–Beltrami operator on $\mathbb{S}^{d-1}$. Writing it in the standard
spherical coordinates $\vartheta_{1},\dots,\vartheta_{d}$ one obtains (see,
e.g., equation (2.2.4) in [KMR])

(4.11)
$\Delta=\sum_{j=1}^{d-1}\frac{1}{q_{j}(\sin\vartheta_{j})^{d-1-j}}\frac{\partial}{\partial\vartheta_{j}}\left((\sin\vartheta_{j})^{d-1-j}\frac{\partial}{\partial\vartheta_{j}}\right),$

where $q_{1}=1$ and $q_{j}=(\sin\vartheta_{1}\dots\sin\vartheta_{j-1})^{2}$
for $j>1$.

For a fixed $y\in\mathbb{S}^{d-1}$, choose the coordinates so that
$\cos\vartheta_{1}=\langle y,x\rangle$. Then $\langle
y,x\rangle^{p}=\cos^{p}\vartheta_{1}$, effectively leaving just one term in
the formula above, and a direct computation shows that

(4.12) $\Delta_{x}\left(\langle x,y\rangle^{p}\right)=p(p-1)\langle
x,y\rangle^{p-2}-p(p+d-1)\langle x,y\rangle^{p}.$

Observe that if $p\in(0,1]$, then the operator $\Delta_{x}$ satisfies
conditions (i) and (ii), hence completing the proof for this range of $p$.

Now consider the operator $D=\Delta\,\big{(}\Delta+p(p+d-1)\big{)}$. It is
easy to see that

(4.13)
$\displaystyle\Delta_{x}\,\big{(}\Delta_{x}+p(p+d-1)\big{)}\left(\langle
x,y\rangle^{p}\right)$ $\displaystyle=p(p-1)\Delta_{x}\left(\langle
x,y\rangle^{p-2}\right)$ $\displaystyle=p(p-1)(p-2)\langle
x,y\rangle^{p-4}\cdot\big{(}(p-3)-(p+d-3)\langle x,y\rangle^{2}\big{)}.$

If $p\in(2,3]$, then $p-3\leq 0$ and $p+d-3>d-1\geq 0$, so the expression
above is strictly negative. Hence this operator satisfies conditions (i) and
(ii) for $2<p\leq 3$.

Moreover, if $p\in(1,2)$, the expression above is strictly positive. Indeed,
the function $g_{p}(t)=(p-3)-(p+d-3)t$ is monotone on $[0,1]$ with
$g_{p}(0)=p-3<0$ and $g_{p}(1)=-d<0$. Therefore, condition (ii) holds with an
opposite inequality sign, so the case $1<p<2$ is also covered.

It is now clear how to iterate this process. Define now the operator
$D^{(0)}=\Delta$, $D^{(1)}=\Delta\,\big{(}\Delta+p(p+d-1)\big{)}$, and, more
generally, for $k\in\mathbb{N}$, define the differential operator of order
$2k+2$

(4.14)
$D^{(k)}=\Delta\left(\Delta+(p+d-2k+1)\prod_{j=0}^{2k-2}(p-j)\right)\cdots\Bigg{(}\Delta+p(p-1)(p-2)(p+d-3)\Bigg{)}\big{(}\Delta+p(p+d-1)\big{)}.$

Let $p\in\mathbb{R}_{+}\setminus 2\mathbb{N}$ and choose $k\in\mathbb{N}_{0}$
so that $2k-1<p\leq 2k+1$. An iterative computation shows that

(4.15) $\displaystyle D^{(k)}_{x}\left(\langle x,y\rangle^{p}\right)$
$\displaystyle=\left(\prod_{j=0}^{2k+1}(p-j)\right)\,\langle
x,y\rangle^{p-2k-2}-\left(\prod_{j=0}^{2k}(p-j)\right)(p+d-2k-1)\,\langle
x,y\rangle^{p-2k}$ $\displaystyle=\left(\prod_{j=0}^{2k}(p-j)\right)\,\langle
x,y\rangle^{p-2k-2}\cdot\left((p-2k-1)-(p+d-2k-1)\,\langle
x,y\rangle^{2}\right).$

For $p\in(2k,2k+1]$, the expression above is strictly negative, since
$p-2k-1\leq 0$ and $p+d-2k-1>d-1\geq 0$.

At the same time, for $p\in(2k-1,2k)$, this expression is strictly positive,
because $\prod_{j=0}^{2k}(p-j)<0$ and the monotone function
$g_{p}(t)=(p-2k-1)-(p+d-2k-1)t$ takes values $g_{p}(0)=p-2k-1<0$ and
$g_{p}(1)=-d<0$. Thus, operator $D^{(k)}$ allows us to prove Propostion 4.3
for $p$ in the range $(2k-1,2k)\cup(2k,2k+1]$. ∎

We suspect that an analog of Theorem 1.3 also holds for the Fejes Tóth
conjecture [FT] mentioned in the introduction. Recall that this conjecture
(its continuous version) deals with the energy $I_{f}$ with potential
$f(t)=\arcsin|t|$ and speculates that the discrete measure uniformly
concentrated on the elements of an orthonormal basis minimizes $I_{f}$. If the
conjecture is true, not all the minimizers of this energy are discrete. For
example, as observed in [BM], on $\mathbb{S}^{3}$, normalized uniform
$1$-dimensional Hausdorff measure on two orthogonal copies of
$\mathbb{S}^{1}$, i.e. on the set

$\\{(x_{1},x_{2},0,0):\,x_{1}^{2}+x_{2}^{2}=1\\}\cup\\{(0,0,x_{3},x_{4}):\,x_{3}^{2}+x_{4}^{2}=1\\},$

would also yield a minimizer. This effect is related to the fact that, while
the kernel $f(t)=\arcsin|t|$ is not positive definite on $\mathbb{S}^{d-1}$
with $d\geq 3$, it is indeed positive definite on $\mathbb{S}^{1}$, i.e. the
uniform measure is a minimizer on the circle. Thus, assuming the conjecture,
this energy does have non-discrete minimizers.

##  5\. Minimizers of energies with analytic kernels

We can also prove a statement analogous to Theorem 1.3 for a wide class of
energies – namely, those with analytic potentials.

######  Theorem 5.1.

Assume that $f$ is a real-analytic function on $[-1,1]$, such that $\sigma$ is
not a minimizer of $I_{f}$, i.e. $f$ is not (up to an additive constant)
positive definite on $\mathbb{S}^{d-1}$. Let $\mu$ be a minimizer of $I_{f}$,
then $(\operatorname{supp}(\mu))^{\circ}=\emptyset$. Moreover, when $d=2$,
then $\operatorname{supp}(\mu)$ must be discrete

###### Proof.

Suppose, indirectly, that $(\operatorname{supp}(\mu))^{\circ}\neq\emptyset$.
By Lemma 4.1, we know that the potential

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

is constant on $\operatorname{supp}(\mu)$. Since $f(\langle x,y\rangle)$ is
real-analytic on $\mathbb{S}^{d-1}\times\mathbb{S}^{d-1}$, $F_{\mu}(x)$ is
real-analytic on $\mathbb{S}^{d-1}$. Since $F_{\mu}$ is real-analytic and
constant on an open set in $\mathbb{S}^{d-1}$, it is constant on all of
$\mathbb{S}^{d-1}$ [Q, Lemma 2.4]. In addition, $F_{\sigma}(x)=I_{f}(\sigma)$
is constant on $\mathbb{S}^{d-1}$ due to rotational invariance. We then obtain

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

This is clearly a contradiction, as by the assumption, $I_{f}$ is not
minimized by $\sigma$. Our first claim then follows.

For $\mathbb{S}^{1}$, we have that if $F_{\mu}$ is constant on a set
$\\{z_{1},z_{2},...\\}\subset\mathbb{S}^{1}$ with an accumulation point,
$F_{\mu}$ is constant on $\mathbb{S}^{1}$. The proof of our second claim then
follows as above. ∎

If $\mathbb{S}^{d-1}$ is replaced with one of the projective spaces
$\mathbb{FP}^{d-1}$ ($\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$, for instance) a
similar result can be derived as above. In this case kernels $f$ are also
functions of the cosine of the geodesic distance $\tau(x,y)=2|\langle
x,y\rangle|^{2}-1$ under identification of points with unit vectors
$x,y\in\mathbb{F}^{d}$; see [BGM+] for more details on energy integrals over
these spaces.

In the spirit of Theorem 5.1, as well as Corollary 2.5, it may be tempting to
conjecture that if $f$ (not necessarily analytic) is not positive definite on
$\mathbb{S}^{d-1}$ (up to constant), i.e $I_{f}(\mu)$ is not minimized by
$\sigma$, then the support of any minimizer of $I_{f}$ must have empty
interior. However, this is not true, as the following simple example shows.
Assume that $f\in C[-1,1]$ is constant near $t=1$ and strictly decreasing
otherwise, i.e. it satisfies for some fixed $\gamma\in(0,1)$,

$f(1)=f(\tau)=\min_{t\in[-1,1]}f(t)\,\,\textup{ for any
}\,\,\tau\in[1-\gamma,1],$

and $f(\tau)>f(1)$ for all $\tau\in[-1,1-\gamma)$. It is then evident that for
any $z\in\mathbb{S}^{d-1}$

$\min_{\mu\in\mathcal{P}(\mathbb{S}^{d-1})}I_{f}(\mu)=I_{f}\big{(}\delta_{z}\big{)}=f(1),$

and $I_{f}(\sigma)>I_{f}(\delta_{x})$, i.e. $\sigma$ is not a minimizer of
$I_{f}$. Let $C(z,\alpha)=\\{x\in\mathbb{S}^{d-1}:\,\langle
x,z\rangle>\alpha\\}$ denote the spherical cap of “height” $\alpha$ centered
at $z\in S^{d-1}$. Let $\nu$ be the normalized uniform measure on
$C(z,\alpha)$, i.e.

$d\nu(x)=\frac{{\bf{1}}_{C(z,\alpha)}(x)}{\sigma\big{(}C(z,\alpha)\big{)}}d\sigma(x),$

with $\alpha=1-\frac{\gamma}{4}$. Then for each $x,y\in C(z,\alpha)$, we have
$\langle x,y\rangle>1-\gamma$, and hence

$I_{f}(\nu)=I_{f}\big{(}\delta_{z}\big{)}=f(1),$

i.e. $\nu$ is also a minimizer of $I_{f}$, but its support has non-empty
interior.

##  6\. Applications of the results to energies with polynomial kernels

We observe that the results of Sections 3 and 5 apply if $f$ is a polynomial.
Indeed, Theorem 5.1 is applicable since polynomials are analytic, while the
conditions of Theorem 3.3 hold because the Gegenbauer expansion has only
finitely many terms. We summarize these statements in the following corollary.

######  Corollary 6.1.

Assume that $f$ is a polynomial whose Gegenbauer expansion is

(6.1) $f(t)=\sum_{n=0}^{m}a_{n}C_{n}^{\lambda}(t).$

  1. (i)

There exists a discrete minimizer $\mu\in\mathcal{P}(\mathbb{S}^{d-1})$ with

$\big{|}\operatorname{supp}\mu\big{|}\leq 1+\sum_{\begin{subarray}{c}\\{n:\
a_{n}>0,\ 1\leq n\leq
m\\}\end{subarray}}\operatorname{dim}\mathcal{H}_{n}^{d}.$

  2. (ii)

If, moreover, $\sigma$ is not a minimizer of $I_{f}$ over
$\mathcal{P}(\mathbb{S}^{d-1})$, i.e. there exists $n\geq 1$ such that
$a_{n}<0$, then the support of any minimizer of $I_{f}$ has empty interior.
For $\mathbb{S}^{1}$ the support is finite.

We observe that when $a_{n}>0$ for $n=1,\dots,m$, i.e. $f$ is positive
definite on $\mathbb{S}^{d-1}$ polynomial (up to constant), the statement of
Theorem 3.3 (and hence also part (i) of the above corollary) is well known. In
this case, the discrete minimizers $\mu=\sum\omega_{x_{i}}\delta_{x_{i}}$ are
exactly weighted spherical $m$-designs, i.e. for any polynomial $P$ of degree
at most $m$ we have

$\sum_{i}\omega_{x_{i}}P(x_{i})=\int_{\mathbb{S}^{d-1}}P(x)d\sigma(x).$

A certain well-known generalization of this fact can also be easily deduced
from part (i) of Corollary 6.1. Let $M\subset\mathbb{N}_{0}$ with $0\in M$.
Call a set $\\{x_{i}\\}_{i=1}^{k}\subset\mathbb{S}^{d-1}$ with positive
weights $\omega_{x_{i}}$ a weighted $\mathcal{M}$-design if for every
$m\in\mathcal{M}$ and for every spherical harmonic $Y\in\mathcal{H}_{m}^{d}$
one has

$\sum_{i}\omega_{x_{i}}Y(x_{i})=\int_{\mathbb{S}^{d-1}}Y(x)d\sigma(x).$

When $\mathcal{M}=\\{0,1,\dots,m\\}$, this definition coincides with the
definition of an $m$-design. Such objects arise naturally for some
configurations. For example, the $600$-cell, one of the six $4$-dimensional
convex regular polytopes with vertices which form a $120$-point subset of
$\mathbb{S}^{3}$, yields an exact cubature formula for spherical harmonics of
degrees up to $19$, excluding degree $12$. In other words, it is an
$\mathcal{M}$-design for
$\mathcal{M}=\\{0,1,\dots,11\\}\cup\\{13,\dots,19\\}$. By taking $a_{n}>0$
only for $n\in\mathcal{M}$ and applying part (i) of Corollary 6.1, one easily
concludes existence of weighted $\mathcal{M}$-designs on the sphere
$\mathbb{S}^{d-1}$ of cardinality at most
$\sum_{n\in\mathcal{M}}\operatorname{dim}\mathcal{H}_{n}^{d}$. This statement
is encompassed by more general results [T, Ro2]

Theorem 3.3 and part (i) of Corollary 6.1 vastly generalize these well-known
statements, essentially showing that the addition of any number of negative
definite terms does not destroy the statement: discrete minimizers with the
same cardinality still exist.

Concerning part (ii) of Corollary 6.1, it might be interesting to give some
explicit examples of polynomials $f$ with at least one negative coefficient
$a_{n}<0$ for $n\geq 1$, for which the minimizers of $I_{f}$ are not
necessarily discrete. Finally, we mention that the case of energy optimization
for polynomial potentials in $d=2$ is more approachable than in higher
dimensions, due to the classical solution of the trigonometric moment problem
[ST, Theorem 1.4].

##  7\. Local minimizers of the $p$-frame energy with $p\in 2\mathbb{N}$ are
global.

Finally, we make an observation that for energies with positive definite
kernels, including the $p$-frame energy with $p\in 2\mathbb{N}$, every local
minimizer is necessarily global. We consider local minima in a rather general
sense.

######  Definition 7.1.

We say that a probability measure $\xi\in\mathcal{P}(\mathbb{S}^{d-1})$ is a
local minimizer of $I_{f}$ if for each $\mu\in\mathcal{P}(\mathbb{S}^{d-1})$
and for any $\tau>0$ small enough (depending on $\mu$),

$I_{f}\big{(}\xi\big{)}\leq I_{f}\big{(}(1-\tau)\xi+\tau\mu\big{)}.$

Observe that this definition is satisfied if $\xi$ is a local minimum with
respect to many reasonable metrics on $\mathcal{P}(\mathbb{S}^{d-1})$, i.e. if
there exists $\varepsilon>0$ such that $I_{f}(\xi)\leq I_{f}(\mu)$ whenever
$d(\xi,\mu)<\varepsilon$, where $d(\xi,\mu)$, represents, for example, the
$d_{p}$-Wasserstein distance, $p<\infty$, or the total variation distance
between measures. The following proposition provides a relation between the
local and global minimizers.

######  Proposition 7.2.

Let $f\in C[-1,1]$ and let $\nu\in\mathcal{P}(\mathbb{S}^{d-1})$ be a global
minimizer of $I_{f}$. Assume also that $\xi\in\mathcal{P}(\mathbb{S}^{d-1})$
is a local minimizer of $I_{f}$ and that
$\operatorname{supp}\xi\subset\operatorname{supp}\nu$. Then $\xi$ is also a
global minimizer of $I_{f}$ over $\mathcal{P}(\mathbb{S}^{d-1})$.

If the function $f$ is positive definite (modulo a constant term) on the
sphere $\mathbb{S}^{d-1}$, then the uniform measure $\sigma$ minimizes $I_{f}$
according to part (iv) of Proposition 2.2, hence one can take $\nu=\sigma$ in
the lemma above. Since $\sigma$ is supported on the whole sphere, this
immediately leads to non-existence of local minimizers which are not global:

######  Corollary 7.3.

Let $f\in C[-1,1]$ be positive definite on $\mathbb{S}^{d-1}$ (up to an
additive constant) and let $\xi$ be a local minimizer of $I_{f}$. Then $\xi$
is necessarily a global minimizer of $I_{f}$, i.e.

$I_{f}(\xi)=\min_{\mu\in\mathcal{P}(\mathbb{S}^{d-1})}I_{f}(\mu).$

###### Proof of Proposition 7.2.

Let $\nu$ be a global minimizer, that is

$I_{f}(\nu)=\inf\limits_{\mu\in\mathcal{P}(\mathbb{S}^{d-1})}I_{f}(\mu)=\alpha.$

According to Lemma 4.1, the potential of $\nu$ satisfies

(7.1) $F_{\nu}(x)=\int_{\mathbb{S}^{d-1}}f(\langle
x,y\rangle)d\nu(y)=I_{f}(\nu)=\alpha\,\,\,\quad\textup{for all
$x\in\operatorname{supp}\nu$.}$

Suppose, by contradiction, that $\xi$ satisfies
$\alpha=I_{f}(\nu)<I_{f}(\xi)$. Since $\xi$ is a local minimizer, setting
$\mu_{\tau}=\tau\nu+(1-\tau)\xi$, for sufficiently small $0<\tau<1$, we have

(7.2) $I_{f}(\mu_{\tau})\geq I_{f}(\xi).$

Setting $I_{f}(\xi)=\beta>\alpha$ and using (7.1), a quick calculation shows
that

$\displaystyle I_{f}(\xi)$ $\displaystyle\leq
I_{f}(\mu_{\tau})=\tau^{2}I_{f}(\nu)+(1-\tau)^{2}I_{f}(\xi)+2\tau(1-\tau)\int_{\mathbb{S}^{d-1}}F_{\nu}(x)\,d\xi(x)$
$\displaystyle=\tau^{2}\alpha+(1-\tau)^{2}\beta+2\tau(1-\tau)\alpha.$

Thus, $\tau^{2}\alpha+2\tau(1-\tau)\alpha+(1-\tau)^{2}\beta\geq\beta$. However

$\tau^{2}\alpha+2\tau(1-\tau)\alpha+(1-\tau)^{2}\beta<\beta\big{(}\tau^{2}+2\tau(1-\tau)+(1-\tau)^{2}\big{)}=\beta,$

which is a contradiction. ∎

Corollary 7.3 obviously applies to the $p$-frame energies when $p=2k$ is an
even integer. As discussed in the introduction, $\sigma$ minimizes $I_{f}$,
since $f(t)=t^{2k}$ is positive definite. Thus, all the local minimizers of
the $2k$-frame energy are necessarily global. A somewhat similar effect for
$p=2$ has been observed in [BeF] for discrete energies: it was proved that any
finite configuration locally minimizing the $N$-point frame energy is also a
global minimizer, and therefore it is a tight frame, whenever $N\geq d$.

##  8\. Acknowledgments

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 paper is based upon work supported by the
National Science Foundation grant DMS-1439786 while D.B., A.G., and O.V. were
in residence at the Institute for Computational and Experimental Research in
Mathematics in Providence, RI, during the “Point Configurations in Geometry,
Physics and Computer Science” program. This work is also supported in part by
EPSRC grant no EP/K032208/1.

We thank Henry Cohn, David de Laat, Axel Flinth, Michael Lacey, Svitlana
Mayboroda, Bruno Poggi, Alexander Reznikov, Daniela Schiefeneder, Alexander
Shapiro, and Yao Yao for helpful discussions.

## References

  * [BCL+] D. Balagué, J. Carrillo, T. Laurent, and G. Raoul.  Nonlocal interactions by repulsive–attractive potentials: radial ins/stability. Phys. D 260 (2013), 5–25. arXiv:1109.5258 MR3143991
  * [BFSM] L. Bäuml, F. Finster, D. Schiefeneder, and H. von der Mosel.  Singular Support of Minimizers of the Causal Variational Principle on the Sphere. arXiv:1808.09754
  * [BeF] J. J. Benedetto and M. Fickus.  Finite normalized tight frames. Adv. Comput. Math, 18 (2003), 357–385. MR1968126
  * [BiD] D. Bilyk and F. Dai.  Geodesic distance Riesz energy on the sphere.  Trans. Amer. Math. Soc., 372, (2019) 3141–3166. arXiv:1612.08442 MR3988605
  * [BGM+] D. Bilyk, A. Glazyrin, R. Matzke, J. Park, and O. Vlasiuk.  Optimal measures for $p$-frame energies on spheres.  pre-print. arXiv:1908.00885
  * [BM] D. Bilyk and R. W. Matzke.  On the Fejes Tóth problem about the sum of angles between lines. Proc. Amer. Math. Soc. 147 (2019), 51–59. arXiv:1801.07837 MR3876730
  * [Bj] G. Bjorck.  Distributions of positive mass, which maximize a certain generalized energy integral.  Ark. för Mat. 3 (1956), 255–269. MR0078470
  * [BHS] S. Borodachov, D. Hardin, and E. Saff.  Minimal Discrete Energy on Rectifiable Sets.  To appear, Springer.
  * [BCD+] C. Boyer, A. Chambolle, Y. De Castro, V. Duval, F. de Gournay, and P. Weiss.  On representer theorems and convex regularization. SIAM J. Optim. 29 (2019), 1260–1281. arXiv:1806.09810 MR3948246
  * [CFP] J. Carrillo, A. Figalli, and F. S. Patacchini.  Geometry of minimizers for the interaction energy with mildly repulsive potentials. Ann. Inst. H. Poincaré Anal. Non Linéaire, 34 (2017), 1299–1308. arXiv:1607.08660 MR3742525
  * [CMV] J. A. Carrillo, R. J. McCann, and C. Villani.  Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates. Rev. Mat. Iberoamericana 19 (2003), 971–1018. MR2053570
  * [CRPW] V. Chandrasekaran, B. Recht, P. Parrilo, and A. Willsky.  The convex geometry of linear inverse problems. Found. Comput. Math. 12 (2012), 805–849. arXiv:1012.0621 MR2989474
  * [DX] F. Dai and Y. Xu.  Approximation Theory and Harmonic Analysis on Spheres and Balls.  Springer Monographs in Mathematics, Springer, New York (2013). MR3060033
  * [DGS] P. Delsarte, J. M. Goethals, and J. J. Seidel.  Spherical codes and designs. Geometriae Dedicata, 6 (1977), 363–388. MR0485471
  * [D] R. G. Douglas.  On Extremal Measures and Subspace Density.  Michigan Math. J. 11 (1964), 243–246. MR0185427
  * [EO] M. Ehler and K. A. Okoudjou.  Minimization of the probabilistic p-frame potential.  J. Statist. Plann. Inference, 142 (2012), 645–659. arXiv:1101.0140 MR2853573
  * [FT] L. Fejes Tóth.  Über eine Punktverteilung auf der Kugel. Acta Math. Acad. Sci. Hungar. 10 (1959), 13–19. (German, Russian summary). MR0105654
  * [FS] F. Finster and D. Schiefeneder.  On the Support of Minimizers of Causal Variational Principles.  Arch. Ration. Mech. Anal. 210 (2013), 321–364. arXiv:1012.1589 MR3101787
  * [FH] C. H. FitzGerald and R. A. Horn.  On Fractional Hadamard Powers of Positive Definite Matrices.  J. Math. Anal. Appl. 61 (1977), 47–50. MR0506356
  * [G] R. Gangolli.  Positive definite kernels on homogeneous spaces and certain stochastic processes related to Lévy’s Brownian motion of several parameters.  Ann. Inst. H. Poincaré Sect. B (N.S.), 3 (1967), 121–226. MR0215331
  * [KKLS] K. Kang, H. K. Kim, T. Lim, and G. Seo.  Uniqueness and characterization of local minimizers for the interaction energy with mildly repulsive potentials.  pre-print. arXiv:1907.07004
  * [K] A. F. Karr.  Extreme Points of Certain Sets of Probability Measures, with Applications.  Math. Oper. Res. 8 (1983), 74–85. MR0703827
  * [KSU+] T. Kolokolnikov, H. Sun, D. Uminsky, and A. L. Bertozzi.  Stability of ring patterns arising from two-dimensional particle interactions.  Phys. E 84 (2011).
  * [KMR] V. Kozlov, V. G. Maz’ya, and J. Rossmann.  Spectral Problems Associated with Corner Singularities of Solutions of Elliptic Equations.  Mathematical surveys and monographs, 85, American Mathematical Society, Providence, RI, (2001). MR1788991
  * [Kn] D. E. Knuth.  The art of computer programming. Vol. 1. Fundamental algorithms. Third edition. Addison-Wesley, Reading, MA, (1997). MR3077152
  * [L] N.S. Landkof.  Foundations of Modern Potential Theory.  Translated from the Russian by A. P. Doohovskoy.  Die Grundlehren der mathematischen Wissenschaften, Band 180, Springer-Verlag, New York-Heidelberg, (1972). MR0350027
  * [LS] P. W. H. Lemmens and J. J. Seidel.  Equiangular lines.  J. Algebra, 24 (1973), 494–512.
  * [LM] T. Lim and R. J. McCann.  Isodiametry, variance, and regular simplices from particle interactions.  pre-print. arXiv:1907.13593
  * [Ma] A. A. Makhnev.  On the nonexistence of strongly regular graphs with the parameters $(486,165,36,66)$ Ukraïn. Mat. Zh., 54 (2002), 941–949. MR2015515
  * [Me] J. Mercer.  Functions of positive and negative type and their connection with the theory of integral equations. Philos. Trans. Roy. Soc. A, 209 (1909), 415–446.
  * [MEB+] A. Mogilner, L. Edelstein-Keshet, L. Bent, and A. Spiros.  Mutual interactions, potentials, and individual distance in a social aggregation. J. Math. Biol. 47 (2003), 353–389. MR2024502
  * [Mu] C. Müller.  Spherical Harmonics.  Lecture Notes in Mathematics, 17, Springer-Verlag, Berlin-New York, (1966). MR0199449
  * [PS] G. Pólya and G. Szegő.  Problems and theorems in analysis, Vol. II.  Revised and enlarged English translation by C. E. Billigheimer, Springer-Verlag, New York-Heidelberg, (1976). MR465631
  * [Q] E. T. Quinto.  Mean value extension theorems and microlocal analysis. Proc. Amer. Math. Soc. 131 (2003), 3267–3274. MR1992868
  * [RBSC] J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves.  Symmetric informationally complete quantum measurements. J. Math. Phys., 45 (2004), 2171–2180. arXiv:quant-ph/0310075 MR2059685
  * [Ri] H. Richter.  Parameterfreie Abschätzung und Realisierung von Erwartungswerten. (German) Bl. Deutsch. Ges. Versicherungsmath. 3  (1957), 147–162. MR0088130
  * [Ro1] W. W. Rogosinski.  Moments of non-negative mass. Proc. Roy. Soc. London Ser. A 245 (1958), 1–27. MR0149217
  * [Ro2] W. W. Rogosinski.  Non-negative linear functionals, moment problems, and extremum problems in polynomial spaces. Studies in mathematical analysis and related topics, Stanford Univ. Press, Stanford, Calif, (1962), 316–324. MR0146643
  * [Ros] P. C. Rosenbloom.  Quelques classes de problèmes extrémaux.  Bull. Soc. Math. France 79 (1951), 1–58. MR0057463
  * [Sch] I. J. Schoenberg.  Positive definite functions on spheres.  Duke Math. J., 9 (1941), 96–108. MR0005922
  * [SG] A. J. Scott and M. Grassl.  Symmetric informationally complete positive- operator-valued measures: a new computer study.  J. Math. Phys., 51 (2010). arXiv:0910.5784 MR2662471
  * [Sh] O. Shatalov.  Isometric Embeddings $l_{2}^{m}\to l_{p}^{n}$ and Cubature Formulas Over Classical Fields.  Ph.D. thesis, Technion–Israel Institute of Technology, (2001).
  * [ST] J. A. Shohat and J. D. Tamarkin.  The Problem of Moments.  American Mathematical Society Mathematical surveys, vol. I. American Mathematical Society, New York, (1943). MR0008438
  * [Si] V. M. Sidel’nikov.  New estimates for the closest packing of spheres in $n$-dimensional Euclidean space.  Mat. Sb., 24 (1974), 148–158. MR0362060
  * [Sz] G. Szegő.  Orthogonal Polynomials.  American Mathematical Society, Colloquium Publications, Vol. XXIII. American Mathematical Society, Providence, R.I., (1975). MR0372517
  * [T] V. Tchakaloff.  Formules de cubatures mécaniques à coefficients non négatifs.  Bull. Sci. Math. 81 (1957), 123–134. MR0094632
  * [UFW] M. Unser, J. Fageot, and J. P. Ward.  Splines are universal solutions of linear inverse problems with generalized TV regularization. SIAM Rev. 59 (2017), 769–793. arXiv:1603.01427 MR3720356
  * [VUK+] J. H. Von Brecht, D. Uminsky, T. Kolokolnikov, and A. L. Bertozzi.  Predicting pattern formation in particle interactions.  Math. Models Methods Appl. Sci. 22 (2012). MR2974182
  * [W] G. Winkler.  Extreme points of moment sets. Math. Oper. Res. 13 (1988), 581–587. MR0971911
  * [WS] L. Wu and D. Slepčev.  Nonlocal interaction equations in environments with heterogeneities and boundaries. Comm. Partial Differential Equations 40 (2015), 1241–1281. MR3341204
  * [Z] G. Zauner.  Grundzüge einer nichtkommutativen Designtheorie. PhD thesis, University of Vienna, 1999.  Published in English translation: Int. J. Quantum Inf. 9 (2011), 445–507. MR2931102
  * [Zu] S. Zuhovickiĭ.  On the approximation of real functions in the sense of P. L. Čebyšev. Amer. Math. Soc. Transl. 19 (1962), 221–252. MR0131706