[1]Cameron Davies

[1]Department of Mathematics, University of Toronto, 40 St. George St.,
Toronto, M5S 2E4, Ontario, Canada

# Bounds and Limiting Minimizers for a Family of Interaction Energies

<EMAIL_ADDRESS>*

###### Abstract

We study a two parameter family of energy minimization problems for
interaction energies $\mathcal{E}_{\alpha,\beta}$ with attractive-repulsive
potential $W_{\alpha,\beta}$. We develop a concavity principle, which allows
us to provide a lower bound on $\mathcal{E}_{\alpha,\beta}$ if there exist
$\beta_{0}<\beta<\beta_{1}$ with minimizers of
$\mathcal{E}_{\alpha,\beta_{0}}$ and $\mathcal{E}_{\alpha,\beta_{1}}$ known.
In addition to this, we also derive new conclusions about the limiting
behaviour of $\mathcal{E}_{\alpha,\beta}$ for $\beta\approx 2.$ Finally, we
describe a method to show that, for certain values of $(\alpha,\beta),$
$\mathcal{E}_{\alpha,\beta}$ cannot be minimized by the uniform distribution
over a top-dimensional regular unit simplex. Our results are made possible by
two key factors – recent progress in identifying minimizers of
$\mathcal{E}_{\alpha,\beta}$ for a range of $\alpha$ and $\beta$, and an
analysis of $\inf\mathcal{E}_{\alpha,\beta}$ as a function on parameter space.

###### keywords:

Energy minimization, aggregation equation, attractive-repulsive power-law
interaction, convex, concave interpolation, weak convergence of measures.

©2023 by the author. The author is grateful to Robert McCann, Ryan Matzke,
Almut Burchard, Doug Hardin, Ihsan Topaloglu, and Rupert Frank for stimulating
interactions. The author acknowledges partial support of his research by a
Canada Graduate Scholarship - Doctoral Natural Sciences and Engineering
Research Council of Canada. The author has no conflicts of interest to report.

##  1 Introduction

The purpose of this paper is to study minimizers of certain interaction energy
functionals $\mathcal{E}_{W}:\mathcal{P}(\mathbb{R}^{n})\to\mathbb{R}$ of the
form

$\mathcal{E}_{W}[\mu]=\frac{1}{2}\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}W(x-y)d\mu(x)d\mu(y),$
(1.1)

with interaction kernel $W:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}$.
More concretely, we study the two parameter family of potentials

$W_{\alpha,\beta}(x-y)=\frac{|x-y|^{\alpha}}{\alpha}-\frac{|x-y|^{\beta}}{\beta},$
(1.2)

for $\alpha>\beta>-n,$ and for such potentials we will use the notation
$\mathcal{E}_{\alpha,\beta}:=\mathcal{E}_{W_{\alpha,\beta}}.$

Energies of the form (1.1) are closely related to the dynamics of particles
whose pairwise interactions are governed by $W.$ More specifically, it is
well-known that the $d_{2}$-gradient flow of $\mathcal{E}_{W}$ is the
aggregation equation

$\frac{\partial\mu}{\partial t}=\nabla\cdot(\mu\nabla W*\mu)$ (1.3)

(see, for example, [1, Section 11.2.1] or [2, Section 8.4.2]). Potential
functions in the form (1.2) are called attractive-repulsive power-law
potentials, and have the benefit of providing a relatively simple, yet still
intriguing, model of particle dynamics where particles which are far apart
attract each other, whereas particles which are close together repel each
other. We will review the literature on the interaction energy (1.1) with
potentials of the form (1.2) later in the introduction. For the time being, we
note that attractive-repulsive particle models have been an attractive choice
across many disciplines, from collective animal behaviour, to granular media
in physics, to game theory, which I describe in more detail elsewhere in [3,
Section 1.3].

We will focus on models of this behaviour which are based on isotropic power
law potentials. However, we also remark on a few other similar problems which
have received a lot of research interest. In particular, the case of
anisotropic potentials has been particularly popular, and I refer the reader
to the work of Carrillo and Shu in [4] for a review, and some interesting
novel results. Another interesting model of attractive-repulsive dynamics
arises in the Riesz kernel case, where the kernel is taken to be purely
repulsive, but where our goal is to minimize the energy among probability
measures supported on a specified compact set (often a sphere). We refer the
reader to Borodachov, Hardin, and Saff’s book [5] for much of the theory of
Riesz potentials. In particular, [5, Chapter 2] introduces the fundamentals of
the theory. Finally, we should also mention that, in the case of attractive-
repulsive potentials, we can restrict our focus to probability measures whose
density is the indicator function of some set, leading to the minimizing sets
problem studied by Carazzato, Pratelli, and Topaloglu in [6].

###  1.1 Literature Review

We now provide the reader with some background on the study of the aggregation
equation with power-law potentials. A seminal contribution came from Balagué,
Carrillo, Laurent, and Raoul, who in [7] discovered a fundamental distinction
between the cases $\beta>2$ and $\beta<2.$ Namely, in the ‘strongly repulsive’
$\beta<2$ case, nearby particles repel each other strongly enough to prevent
global minimizers of (1.1) from concentrating on sets of zero Hausdorff
dimension, whereas in the ‘mildly repulsive’ $\beta>2$ case, repulsion between
nearby particles is so mild that it is easily overcome by attraction, leading
minimizers of (1.1) to concentrate on sets of Hausdorff dimension $0$. It was
later shown by Carrillo, Figalli, and Patacchini in [8] that global minimizers
for $\mathcal{E}_{\alpha,\beta}$ for $\alpha>\beta>2$ must concentrate on
subsets of $\mathbb{R}^{n}$ with finite cardinality. It should be noted that
neither [7] nor [8] address the case of the centripetal line $\beta=2$. Here,
as we will discuss later in the introduction, the structure of global
minimizers of (1.1) depends strongly on the value of $\alpha.$

In parallel to this study of the properties of abstract global minimizers,
various researchers studied both the linear and non-linear stability of
various steady states of the aggregation equation (1.3), work which would
later inspire myself and others to explicitly identify global minimizers of
(1.1) for certain values of $\alpha$ and $\beta$. Most relevant to this
research, Sun, Uminsky, and Bertozzi investigated the stability of uniform
distributions over spherical shells and found that, for regimes in parameter
space where uniform distributions over spherical shells fail to be stable, the
uniform distribution over the vertices of a top-dimensional unit simplex
appeared to be an attractor for the dynamics in [9]. Kolokolnikov and Von
Brecht also joined this trio to investigate the stability of uniform
distributions over ‘particle rings’, or equally spaced points on circles in
$\mathbb{R}^{2}$ in [10]. Other notable contributions came from Albi, Balagué,
Carrillo, and Von Brecht, who studied particle rings in [11], Balagué,
Carrillo, Laurent, and Raoul, who studied the radial stability of uniform
distributions on spherical shells in [12], and Simione, who studied the
nonlinear stability of simplex configurations in his thesis [13].

Spurred on by both research directions, myself and a number of other
researchers began identifying explicit global minimizers. Since, for
potentials of the form $W_{\alpha,\beta},$ the interaction energy (1.1) is
invariant under translations and rotations of $\mu$, when we say that
minimizers of $\mathcal{E}_{\alpha,\beta}$ are unique, we implicitly mean that
they are unique up to translation and rotation. In general, the following
three types of measures will be especially relevant to our work:

######  Definition 1.1 (Three Useful Measures).

Let $n\geq 1$ and $r\geq 0.$ We define three useful classes of measures as
follows:

  * •

Let $\\{x_{0},\dots,x_{n}\\}$ be points in $\mathbb{R}^{n}$ satisfying
$|x_{i}-x_{j}|=\delta_{ij}$ (i.e. so that they form the vertices of a unit
simplex) such that, for each $i\in\\{0,\dots,n\\},$
$|x_{i}|=\sqrt{\frac{n}{2n+2}}$, the last $n-i-1$ components of each $x_{i}$
are $0,$ and the $i+1$th component of each $x_{i}$ is positive. Then we call
the measure $\mu_{n}^{\Delta}:=\frac{1}{n+1}\sum_{i=1}^{n}\delta_{x_{i}}$ the
centred regular unit $n$-simplex measure, or the unit simplex for short.

  * •

We define the cross-polytope measure of radius $r$ as
$\mu^{*}_{n}(r):=\frac{1}{2n}\sum_{i=1}^{n}(\delta_{re_{i}}+\delta_{-re_{i}}).$

  * •

We define the spherical shell measure of radius $r$ as the uniform
distribution over $\partial B_{r}(0).$

The identification of global minimizers began with the work of Lim and McCann
in [14], who first showed that, for $\beta\geq 2,$ in the hard confinement
limit $\alpha=\infty,$ the minimizers of the interaction
$\mathcal{E}_{\infty,2}$ are uniformly distributed over the vertices of top-
dimensional unit simplices (i.e. maximal collections of points at distance $1$
from each other). They then used $\Gamma$-convergence to show that the same
result holds for $\mathcal{E}_{\alpha,\beta}$ with $\beta\geq 2$ and
sufficiently large $\alpha$ in [14, Theorem 1.3]. Building off of this work,
and making use of the convexity properties of $\mathcal{E}_{4,2}$ discussed by
Lopes in [15], I joined Lim and McCann to explicitly identify and characterize
minimizers of $\mathcal{E}_{\alpha,\beta}$ for a large portion of the mildly
repulsive regime $\alpha>\beta>2$ and the centrifugal line $\alpha>\beta=2$ in
[16, 17]. In particular, because they will be relevant to the results of this
paper, I summarize our contributions from [16, Theorem 1.1], [16, Theorem
1.5], [17, Corollary 2.3], and [17, Theorem 2.4] in the following result:

######  Theorem 1.2 (Minimizers in the Mildly Repulsive Regime and on the
Centrifugal Line from [16, 17]).

Let $\alpha>\beta\geq 2.$ Then:

  * •

If $(\beta,n)=(2,1)$ and $\alpha\geq 3,$ then (1.1) is uniquely minimized by
the unit simplex. Moreover, there exists $\beta_{1}\geq 2$ such that
$\mathcal{E}_{\alpha,\beta}$ is uniquely minimized by the unit simplex for any
$\alpha>\beta\geq\beta_{1}$, and such that there exists a continuous, strictly
decreasing threshold function
$\alpha_{\Delta^{1}}:[2,\beta_{n}]\to[\beta_{n},3]$ with the property that
$\mathcal{E}_{\alpha,\beta}$ is uniquely minimized by the unit simplex for
$\alpha>\alpha_{\Delta^{1}}(\beta),$ and not minimized by the unit simplex for
$\alpha<\alpha_{\Delta^{1}}(\beta).$

  * •

If $n\geq 2$ and $4>\alpha>\beta=2,$ then $\mathcal{E}_{\alpha,\beta}$ is
minimized by a spherical shell of appropriate radius.

  * •

If $n\geq 2$ and $(\alpha,\beta)=(4,2),$ then $\mathcal{E}_{\alpha,\beta}$ is
minimized by any probability measure supported on a sphere of radius
$\sqrt{\frac{n}{2n+2}}$ which satisfies the second moment constraint $\int
x\otimes xd\mu(x)=\frac{1}{2n+2}\operatorname{Id}.$

  * •

If $n\geq 2,$ there exists $\beta_{n}\geq 2$ such that
$\mathcal{E}_{\alpha,\beta}$ is uniquely minimized by the unit simplex for any
$\alpha>\beta\geq\beta_{n}$, and such that there exists a continuous, strictly
decreasing threshold function
$\alpha_{\Delta^{n}}:[2,\beta_{n}]\to[\beta_{n},4]$ with the property that
$\mathcal{E}_{\alpha,\beta}$ is uniquely minimized by the unit simplex for
$\alpha>\alpha_{\Delta^{n}}(\beta),$ and not minimized by the unit simplex for
$\alpha_{\Delta^{n}}(\beta).$

A substantial amount of work has also been undertaken to identify energy
minimizers in the strongly repulsive regime, and along the part of the
centrifugal line which was not addressed by myself, Lim, and McCann. This work
began with Carrillo and Shu, who in [18, Theorem 5.1] and [18, Remark 5.8]
used convexity to show that suitable densities on balls globally minimize
$\mathcal{E}_{2,\beta}$ and $\mathcal{E}_{4,\beta}$ for certain values of
$\beta<2.$ In addition to this, Frank completed the classification of
minimizers on the centripetal line in [19], where he used an elegant argument
based on Fourier analysis to show that, for $n=1$ and $3>\alpha>2,$
$\mathcal{E}_{\alpha,2}$ is uniquely minimized by a certain density on an
interval. Most recently, Frank and Matzke used convexity arguments and a
careful analysis of hypergeometric functions to show that, for a wide range of
parameters $\alpha\in[2,4]$ and for certain $\beta\leq 2,$ minimizers of
$\mathcal{E}_{\alpha,\beta}$ are spherical shells of an appropriate radius
depending on $\alpha$ and $\beta$. We provide a more precise summary of these
results ([18, Theorem 5.1], [18, Remark 5.8], [19, Theorem 1], and [20,
Theorem 1]) below:

######  Theorem 1.3 (Minimizers in the Strongly Repulsive Regime and on the
Centrifugal line from [18, 19, 20]).

The following are true:

  * •

Let $\alpha=2$ and $2-n<\beta<\min(4-n,2),$ or $\alpha=4$ and
$2-n<\beta<\frac{2+2d-d^{2}}{d+1}.$ Then $\mathcal{E}_{\alpha,\beta}$ is
uniquely minimized by a density on a ball of appropriate radius, where the
density has formula given by [18, Equation (5.9)] if $\alpha=2$, or [18,
Equation (5.10)] if $\alpha=4.$

  * •

Let $(\beta,n)=(2,1),$ and let $\alpha\in(2,3).$ Then $\mathcal{E}_{\alpha,2}$
is uniquely minimized by a density on an interval, which are both explicitly
stated in [19, Theorem 1].

  * •

Let $d\geq 2,$ $2\leq\alpha\leq 4,$ and $\frac{-10+3\alpha+7d-\alpha
d-d^{2}}{d+\alpha-3}\leq\beta\leq 2,$ and assume that
$(\alpha,\beta)\neq(4,2).$ Then $\mathcal{E}_{\alpha,\beta}$ is uniquely
minimized by the uniform distribution over the sphere of radius

$R_{\alpha,\beta}=\left(\frac{\Gamma(\frac{d+\beta-1}{2})\Gamma(\frac{2d+\alpha-2}{2})}{\Gamma(\frac{d+\alpha-1}{2})\frac{2d+\beta-2}{2}}\right)^{\frac{1}{\alpha-\beta}}.$

###  1.2 Contributions of this Paper

We now explain the key contributions of the present work. We note that, at a
fundamental level, most of the new results of this paper stem from past work
on identifying global minimizers of $\mathcal{E}_{\alpha,\beta}$ for various
values of $(\alpha,\beta),$ along with a careful study of the minimum value of
$\mathcal{E}_{\alpha,\beta}$ as a function on the space of parameters
$\alpha>\beta>-n$. More precisely, we define the following class of functions:

######  Definition 1.4 (Minimal Energy on a Set of Probability Measures).

Let $\mathcal{F}\subseteq\mathcal{P}(\mathbb{R}^{n})$ be a set of probability
measures. We define the minimal energy of $\mathcal{E}_{\alpha,\beta}$ on
$\mathcal{F}$ by

$E_{\alpha}^{\mathcal{F}}(\beta):=\inf_{\mu\in\mathcal{F}}\mathcal{E}_{\alpha,\beta}[\mu].$

In the case where $\mathcal{F}=\mathcal{P}(\mathbb{R}^{n}),$ we denote
$E_{\alpha}(\beta):=E^{\mathcal{P}(\mathcal{R}^{n})}_{\alpha}(\beta).$

We dedicate Section 2 to studying the minimal energy as a function of
$(\alpha,\beta),$ and exploring the lower bounds on the energy which we can
derive from this perspective. In particular, for $\alpha>0$, this function is
concave in $\beta$ and, if we minimize over all of
$\mathcal{P}(\mathbb{R}^{n}),$ then the minimum is also non-decreasing in
$\beta:$

######  Proposition 1.5 (Properties of the Minimal Energy).

Let $\alpha>0$ be fixed, let $\beta\in(0,\alpha),$ and let
$\mathcal{F}\subseteq\mathcal{P}(\mathbb{R}^{n})$ be a non-empty family of
probability measures. Then $E^{\mathcal{F}}_{\alpha}(\beta)\in\mathbb{R}$ for
every $\beta\in(0,\alpha]$ and, in addition,
$E_{\alpha}^{\mathcal{F}}:(0,\alpha]\to\mathbb{R}$ is a continuous, concave
function. Moreover, $E_{\alpha}$ is non-decreasing on $(0,\alpha].$

In fact, if we fix $0<\beta_{0}<\beta_{1}<\alpha,$ then we can say that, for
$\beta\in[\beta_{0},\beta_{1}],$ $E_{\alpha}$ is strongly concave on
$[\beta_{0},\beta_{1}]$, according to the following definition which can, for
example, be found in [21, Section 2.1.3]:

######  Definition 1.6 (Strong Concavity).

We say that $f:\mathbb{R}\to\mathbb{R}$ is strongly concave on $[a,b]$ with
concavity parameter $m$ if, for all $x_{0},x_{1}\in[a,b],$

$f((1-t)x_{0}+tx_{1})\geq(1-t)f(x_{0})+tf(x_{1})+t(1-t)\frac{m}{2}(x_{0}-x_{1})^{2}.$

If $f$ is $C^{2},$ then it is strongly concave on $[a,b]$ if and only if
$f^{\prime\prime}(x)\geq m$ for all $x\in[a,b].$

If, moreover, we know the minimum value of $\mathcal{E}_{\alpha,\beta_{0}}$
and $\mathcal{E}_{\alpha,\beta_{1}},$ then we can deduce the following lower
bound on $\mathcal{E}_{\alpha,\beta}$ for $\beta\in(\beta_{0},\beta_{1})$ by
concave interpolation:

######  Proposition 1.7 (Strong Concave Interpolation Bound on
$\mathcal{E}_{\alpha,\beta}$).

Let $0<\beta_{0}<\beta_{1}\leq\alpha.$ Then $E_{\alpha}$ is strongly concave
on $[\beta_{0},\beta_{1}],$ with concavity parameter
$\frac{1}{2\beta_{1}^{2}}\frac{n}{n+1}\left[\frac{1}{\alpha}-\frac{1}{\beta_{1}}\right].$
In particular, if $\beta_{t}=(1-t)\beta_{0}+t\beta_{1},$ then

$E_{\alpha}(\beta_{t})\geq(1-t)E_{\alpha}(\beta_{0})+tE_{\alpha}(\beta_{1})+\frac{t(1-t)}{4\beta_{1}^{2}}\frac{n}{n+1}\left[\frac{1}{\alpha}-\frac{1}{\beta_{1}}\right](\beta_{1}-\beta_{0})^{2}.$
(1.4)

In Section 3, we then turn our attention to analyzing the minimizers of
$\mathcal{E}_{\alpha,\beta}$ for $\beta\approx 2.$ When doing so, we recall
that Carrillo, Figalli, and Patacchini showed in [8] that, for $\beta>2,$ all
minimizers of $\mathcal{E}_{\alpha,\beta}$ consist of finite sums of
$\delta$-masses. With this in mind, we introduce the following notation for
spaces of discrete measures with at most $k$ points in their support:

######  Definition 1.8 (Discrete Measures).

Given $k\in\mathbb{N}$ and $\Omega\subseteq\mathbb{R},$ we define the set of
(at most) $k$-point measures
$\mathcal{D}_{k}(\Omega)\subseteq\mathcal{P}(\Omega)$ by

$\mathcal{D}_{k}(\Omega):=\\{\mu\in\mathcal{P}(\Omega)\ |\
|\operatorname{spt}(\mu)|\leq k\\},$

where $|A|$ denotes the cardinality of $A.$

By using Proposition 1.5 and considering these sets of discrete measures, we
are able to show the following result, which states that, if $n=1$ and
$\alpha\in(2,3),$ or if $n\geq 2$ and $\alpha\in(2,4),$ the number of points
in the support of minimizers of $\mathcal{E}_{\alpha,\beta}$ necessarily tends
to $\infty$ as $\beta\searrow 2:$

######  Theorem 1.9 (Cardinality of Energy Minimizers for $\beta\approx 2$).

For $n=1,$ fix $\alpha\in(2,3).$ For $n\geq 2,$ fix $\alpha\in(2,4).$ Then

$\lim_{\beta\to 2^{+}}\min\\{|\operatorname{spt}(\mu)|\ |\
\mu\in\operatorname{argmin}\mathcal{E}_{\alpha,\beta}[\cdot]\\}=+\infty.$

We are also able to use these properties and techniques to say that, in a
certain sense, minimizers of $\mathcal{E}_{\alpha,\beta}$ tend to minimizers
of $\mathcal{E}_{\alpha,2}$ in Theorem 3.6, Proposition 3.8, and Theorem 3.10.

Finally, in Section 4, I continue the work which Lim, McCann, and I started in
[16] on identifying lower bounds for the threshold function
$\alpha_{\Delta^{n}},$ which is defined to be the value such that, for
$\alpha<\alpha_{\Delta^{n}}(\beta),$ unit simplices do not minimize
$\mathcal{E}_{\alpha,\beta}$ and for $\alpha>\alpha_{\Delta^{n}}(\beta),$ unit
simplices uniquely minimize $\mathcal{E}_{\alpha,\beta}.$ In particular, by
comparing the energy of the unit simplex to the energy of the optimal cross-
polytope as defined in Definition 1.1, I define a new family of lower bounds
$\underline{\alpha}_{\Delta^{n}}^{*}$ on $\alpha_{\Delta^{n}}.$ In particular,
these bounds arise from defining a family of unimodal functions $\varphi_{n}$
in Definition 4.4. Once we have done so, and checked a number of properties,
we arrive at the following result:

######  Proposition 1.10 (A Lower Bound for the Transition Threshold).

Fix $\alpha>\beta\geq 2,$ and define
$\underline{\alpha}_{\Delta^{n}}^{*}(\beta)$ as the largest solution of
$\varphi_{n}(\alpha)=\varphi_{n}(\beta).$ Then
$\underline{\alpha}_{\Delta^{n}}^{*}(\beta)\leq\alpha_{\Delta^{n}}(\beta),$
where $\alpha_{\Delta^{n}}(\beta)$ is the simplex transition threshold defined
in [16, Theorem 1.5].

While we have not been able to find a rigorous proof that this bound is
superior to those set out in my joint work with Lim and McCann in [16], I
conclude Section 4, and hence this paper, by providing some evidence which
supports the conclusion that $\underline{\alpha}_{\Delta^{n}}^{*}$ is a better
bound than $\alpha_{\Delta^{n}},$ at least in some cases.

##  2 Parameter Space Concavity and Resulting Bounds

We begin by establishing some basic concavity properties, starting with the
following general proposition, a special case of which provides the basis for
later analysis:

######  Proposition 2.1 (Convexity and Concavity Properties of Parametrized
Energies).

Let $\Phi$ and $\Psi$ be convex parameter spaces and for
$(\varphi,\psi)\in\Phi\times\Psi,$ define
$W_{\varphi,\psi}(x,y)=w_{\varphi}(|x-y|)+w_{\psi}(|x-y|).$ Assume that for
each $(\varphi,\psi)\in\Phi\times\Psi,$ the interaction energy

$\mathcal{E}_{\varphi,\psi}[\mu]:=\frac{1}{2}\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}W_{\varphi,\psi}(x,y)d\mu(x)d\mu(y)$

is uniformly bounded below for
$\mu\in\mathcal{F}\subseteq\mathcal{P}(\mathbb{R}^{d}),$ and define
$E^{\mathcal{F}}_{\psi}:\Phi\to\mathbb{R}$ by
$E^{\mathcal{F}}_{\psi}(\varphi):=\inf_{\mu\in\mathcal{F}}\mathcal{E}_{\varphi,\psi}[\mu].$
Then:

  1. a).

if, for a fixed $\psi\in\Psi,$ the map $\varphi\mapsto w_{\varphi}(t)$ is
concave for each $t\in[0,\infty),$ then for any $\mu\in\mathcal{F},$ the map
$\varphi\mapsto\mathcal{E}_{\varphi,\psi}[\mu]$ is concave on $\Phi$.

  2. b).

if $\psi$ is fixed and the map $\varphi\mapsto\mathcal{E}_{\varphi,\psi}[\mu]$
is concave for each $\mu\in\mathcal{F}$, then the map $\varphi\mapsto
E^{\mathcal{F}}_{\varphi,\psi}$ is concave on $\Phi$.

######  Proof 2.2.

For part a), we fix $t\in[0,1]$ and $\varphi_{0},\varphi_{1}\in\Phi$ and
compute

$\displaystyle\mathcal{E}_{(1-t)\varphi_{0}+t\varphi_{1},\psi_{*}}[\mu]$
$\displaystyle=\frac{1}{2}\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}w_{(1-t)\varphi_{0}+t\varphi_{1}}(|x-y|)+w_{\psi_{*}}(|x-y|)\
d\mu(x)d\mu(y)$
$\displaystyle\geq\frac{1}{2}\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}(1-t)w_{\varphi_{0}}(|x-y|)+tw_{\varphi_{1}}(|x-y|)+w_{\psi_{*}}(|x-y|)\
d\mu(x)d\mu(y)$
$\displaystyle=(1-t)\mathcal{E}_{\varphi_{0},\psi_{*}}[\mu]+t\mathcal{E}_{\varphi_{1},\psi_{*}},$

as desired. Part b) follows from noting that, in this case,
$E^{\mathcal{F}}_{\varphi,\psi_{*}}=\inf_{\mu\in
F}\mathcal{E}_{\varphi},\psi_{*}[\mu]$ is an infimum of concave functions of
$\varphi$ and applying, for example, [22, Theorem 5.5].

This general proposition forms the basis of our proof of Proposition 1.5:

######  Proof 2.3 (Proof of Proposition 1.5).

First note that, for every $\beta\in(0,\alpha],$
$E_{\alpha}^{\mathcal{F}}(\beta)\geq E_{\alpha}(\beta)>-\infty$ by [23,
Theorem 2.3]. We next show that $E_{\alpha}^{\mathcal{F}}$ is concave. This is
an immediate consequence of 2.1 applied with $\Phi=(0,\alpha],$
$\Psi=\\{\alpha\\},$ $w_{\psi}(t)=w_{\alpha}(t)=\frac{t^{\alpha}}{\alpha},$
and, for $\beta\in\Phi,$ $w_{\beta}(t)=-\frac{t^{\beta}}{\beta}.$ In
particular, the assumption that the energy is uniformly bounded below follows
again from [23, Theorem 2.3]. Likewise, concavity of the map $\beta\mapsto
w_{\beta}(t)$ for $t\geq 0$ follows from computing

$-\frac{d^{2}}{d\beta^{2}}\frac{t^{\beta}}{\beta}=\begin{cases}-\frac{t^{\beta}\left((\beta\log
t)^{2}-2(\beta\log t)+2\right)}{\beta^{3}},&t>0\\\ 0,&t=0,\end{cases}$

using the fact $s^{2}-2s+2\geq 1$ for any $s\in\mathbb{R}$ to see that this
second derivative is negative for $t>0.$ Continuity follows from
$E_{\alpha}^{\mathcal{F}}$ being concave through, for example, [22, Corollary
10.1.1].

To prove that $E_{\alpha}^{\mathcal{F}}$ is non-decreasing, first fix
$0<\beta<\beta^{\prime}<\alpha.$ We recall that, from [17, Proposition 2.1],
any global minimizer $\mu$ of $\mathcal{E}_{\alpha,\beta}$ in
$\mathcal{P}(\mathbb{R}^{n})$ satisfies the diameter support bound
$\operatorname{diam}(\operatorname{spt}(\mu))\leq e^{1/\beta}$ and likewise
any global minimizer $\mu$ of $\mathcal{E}_{\alpha,\beta^{\prime}}$ satisfies
$\operatorname{diam}(\operatorname{spt}(\mu))\leq
e^{1/\beta^{\prime}}<e^{1/\beta}.$ As such, we may write

$E_{\alpha}(\beta)=\inf\\{\mathcal{E}_{\alpha,\beta}(\mu)\ |\
\operatorname{diam}(\operatorname{spt}(\mu))\leq e^{1/\beta}\\}.$ (2.1)

and

$E_{\alpha}(\beta^{\prime})=\inf\\{\mathcal{E}_{\alpha,\beta^{\prime}}(\mu)\
|\ \operatorname{diam}(\operatorname{spt}(\mu))\leq e^{1/\beta}\\}.$ (2.2)

Moreover, notice that
$-\frac{d}{d\beta}\frac{t^{\beta}}{\beta}=\frac{t^{\beta}(1-\log(t^{\beta}))}{\beta^{2}}\geq
0$ for $t\in(0,e^{1/\beta}]$ and $\frac{d}{d\beta}\frac{t^{\beta}}{\beta}=0$
if $t=0.$ Thus, the function $-\frac{t^{\beta}}{\beta}$ is non-decreasing in
$\beta$ for $t\in[0,e^{1/\beta}].$ As such, for any $\mu$ with
$\operatorname{diam}(\operatorname{spt}(\mu))\leq e^{1/\beta},$ we find that:

$\displaystyle\mathcal{E}_{\alpha,\beta}[\mu]$
$\displaystyle=\frac{1}{2}\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\frac{|x-y|^{\alpha}}{\alpha}-\frac{|x-y|^{\beta}}{\beta}d\mu(x)d\mu(y)$
$\displaystyle\leq\frac{1}{2}\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\frac{|x-y|^{\alpha}}{\alpha}-\frac{|x-y|^{\beta^{\prime}}}{\beta^{\prime}}d\mu(x)d\mu(y)$
$\displaystyle=\mathcal{E}_{\alpha,\beta^{\prime}}[\mu].$

Taking an infimum over all $\mu\in\mathcal{P}(\mathbb{R}^{n})$ satisfying the
diameter bound, we deduce that $E_{\alpha}(\beta)\leq
E_{\alpha}(\beta^{\prime}),$ as desired.

######  Remark 2.4.

A similar result to Proposition 1.5 holds if we fix $\beta<0$ and consider the
minimal energy as a function of $\alpha\in(\beta,0).$ However, in this regime,
much less is known about minimizers of the interaction energy, so fewer
conclusions can be drawn from this result.

An immediate application of the concavity discussed in Proposition 1.5 is
that, if $\beta_{0},\beta_{1}$ are such that $E_{\alpha}(\beta_{0})$ and
$E_{\alpha}(\beta_{1})$ are known, we may use linear interpolation to find
explicit lower bounds for $E_{\alpha}(\beta)$ for
$\beta\in[\beta_{0},\beta_{1}].$ More precisely, the following holds:

######  Lemma 2.5 (Concave Interpolation Bound on the Energy).

Let $0<\beta_{0}<\beta_{1}\leq\alpha,$ and define
$\beta_{t}:=(1-t)\beta_{0}+t\beta_{1}.$ Then

$E_{\alpha}(\beta_{t})\geq(1-t)E_{\alpha}(\beta_{0})+tE_{\alpha}(\beta_{1}).$

######  Proof 2.6.

This is a direct consequence of Proposition 1.5 and the definition of
concavity.

Of course, it is more insightful to apply the preceding lemma to cases where
minimizers are explicitly known for $\beta_{0}$ and $\beta_{1},$ but not for
$\beta\in(\beta_{0},\beta_{1}).$ However, it is often the case that, even for
values of $(\alpha,\beta)$ with $E_{\alpha}(\beta)$ is explicitly known, the
formula for $E_{\alpha}(\beta)$ can be quite unwieldy (see, for example, [19,
Theorem 1] and [20, Theorem 1]). As such, we instead provide an example
application of Lemma 2.5 as a proof-of-concept, and then provide an overview
of the relevant literature.

######  Example 2.7.

Let $n\geq 2.$ Then for $\alpha\in(2,4),$ and $\beta\in(2,\alpha),$

$E_{\alpha}(\beta)\geq-\frac{\alpha-\beta}{\alpha-2}\frac{2^{d-3}}{\sqrt{\pi}}\frac{\Gamma(\frac{d}{2})\Gamma(\frac{d+\alpha-1}{2})}{\Gamma(\frac{2d+\alpha-2}{2})}\left(\frac{1}{2}-\frac{1}{\alpha}\right)\left(\frac{\Gamma(\frac{d+1}{2})\Gamma(\frac{2d+\alpha-2}{2})}{\Gamma(\frac{d+\alpha-1}{2})\Gamma(d)}\right)^{\frac{\alpha}{\alpha-\beta}}.$

######  Proof 2.8.

Let $\beta_{0}=2$ and $\beta_{1}=\alpha,$ so that
$\beta=\frac{\alpha-\beta}{\alpha-2}\cdot
2+\frac{\beta-2}{\alpha-2}\cdot\alpha.$ By definition
$\mathcal{E}_{\alpha,\alpha}[\mu]=0$ for any
$\mu\in\mathcal{P}(\mathbb{R}^{n})$ so that $E_{\alpha}(\alpha)=0.$ Moreover,
by [20, Theorem 1],

$E_{\alpha}(2)=-\frac{2^{d-3}}{\sqrt{\pi}}\frac{\Gamma(\frac{d}{2})\Gamma(\frac{d+\alpha-1}{2})}{\Gamma(\frac{2d+\alpha-2}{2})}\left(\frac{1}{2}-\frac{1}{\alpha}\right)\left(\frac{\Gamma(\frac{d+1}{2})\Gamma(\frac{2d+\alpha-2}{2})}{\Gamma(\frac{d+\alpha-1}{2})\Gamma(d)}\right)^{\frac{\alpha}{\alpha-\beta}}.$

Thus, by Lemma 2.5, we deduce the desired result.

######  Remark 2.9 (Use Cases of Lemma 2.5).

The bounds provided by Lemma 2.5 are useful in cases where there exist
$\beta_{0}<\beta_{1}$ such that minimizers of $\mathcal{E}_{\alpha,\beta_{0}}$
and $\mathcal{E}_{\alpha,\beta_{1}}$ are known, but minimizers for
$\mathcal{E}_{\alpha,\beta}$ are unknown for at least some
$\beta\in(\beta_{0},\beta_{1}).$ Given the current state of the literature,
Lemma 2.5 can be used to provide energy bounds in the following situations:

  * •

$n=1,$ $\alpha\in(2,3),$ $\beta\in[2,\alpha].$ In this case, the minimal
energy $E_{\alpha}(2)$ is explicitly known due to [19, Theorem 1]. On the
other hand, $E_{\alpha}(\alpha)$ is identically zero, which allows us to
interpolate. However, by applying the work on transition threshold bounds in
[16, Section 4], we can get better bounds for
$\alpha\in\left(\frac{1}{\log(3/2)},3\right).$ In particular, following [16,
Definition 4.1], for such an $\alpha,$ define $\beta_{1}(\alpha)$ as the
smallest solution to

$\frac{(3/2)^{\alpha}}{\alpha}=\frac{(3/2)^{\beta}}{\beta}$

which, in the notation of that paper, ensures that
$\alpha=\alpha^{*}_{\infty}(\beta_{1}(\alpha)).$ Thus, by [16, Corollary 4.5],
$\mathcal{E}_{\alpha,\beta_{1}(\alpha)}$ is uniquely minimized by unit
simplices, so that

$E_{\alpha}(\beta_{1}(\alpha))=\frac{1}{4}\left(\frac{1}{\alpha}-\frac{1}{\beta_{1}(\alpha)}\right)=\frac{1}{4\alpha}(1-(3/2)^{\alpha-\beta_{1}(\alpha)}).$

  * •

$n\geq 2,$ $\alpha\in(2,4),$ $\beta\in[2,\alpha].$ As we discussed in Example
2.7, minimizers of $\mathcal{E}_{\alpha,2}$ for this range of $\alpha$ were
shown to be spherical shells in [17], and $E_{\alpha}(2)$ was calculated in
[20, Theorem 1]. However, analogously to the one-dimensional case, it is
possible to get better bounds if $\alpha\in\left(\frac{2}{\log 2},4\right).$
More specifically, in this case, we can define $\beta_{1}(\alpha)$ as the
smallest solution to $2^{\alpha/2}/\alpha=2^{\beta/2}/\beta,$ which again
allows us to apply [16, Corollary 4.5] to check that
$\mathcal{E}_{\alpha,\beta_{1}(\alpha)}$ is uniquely minimized by unit
simplices, and hence that

$E_{\alpha}(\beta_{1}(\alpha))=\frac{1}{2}\frac{n}{n+1}\left(\frac{1}{\alpha}-\frac{1}{\beta_{1}(\alpha)}\right)=\frac{1}{2\alpha}\frac{n}{n+1}(1-(3/2)^{\alpha-\beta_{1}(\alpha)}).$

We now turn our attention to improving Lemma 2.5 by using strong concavity as
in Definition 1.6. Since it will be useful later, and since we lack a source
for this, we prove the following useful property of strongly concave
functions.

######  Lemma 2.10 (Strong Concavity of the Pointwise Infimum).

Let $\\{f_{i}\\}_{i\in I}$ be a family of strongly concave functions on
$[a,b]$ which are uniformly bounded below on $[a,b]$ and which have uniform
concavity parameter $m.$ Then the pointwise infimum $f:=\inf_{i\in I}f_{i}$ is
also strongly concave with concavity parameter $m.$

######  Proof 2.11.

Fix $x_{0},x_{1}\in[a,b]$ and explicitly compute:

$\displaystyle f((1-t)x_{0}+tx_{1})$ $\displaystyle=\inf_{i\in
I}f_{i}((1-t)x_{0}+tx_{1})$ $\displaystyle\geq\inf_{i\in
I}\left[(1-t)f_{i}(x_{0})+tf_{i}(x_{1})+t(1-t)\frac{m}{2}(x_{0}-x_{1})^{2}\right]$
$\displaystyle\geq(1-t)\inf_{i\in I}f_{i}(x_{0})+t\inf_{i\in
I}f_{i}(x_{1})+t(1-t)\frac{m}{2}(x_{0}-x_{1})^{2}$
$\displaystyle=(1-t)f(x_{0})+tf(x_{1})+t(1-t)\frac{m}{2}(x_{0}-x_{1})^{2},$

as desired.

In the following lemmas, we outline a strategy for proving that
$E_{\alpha}(\beta)$ is strongly concave on certain intervals, note that we can
use it, along with the same techniques described in Example 2.7 and Remark
2.9, to derive lower bounds on the energy. We caution the reader that we make
a number of simplifying estimates which mean that the concavity parameter we
eventually derive is suboptimal. Because of this, we phrase the following
train of thought as a series of short lemmas, rather than a single, longer
result:

######  Lemma 2.12 (Second Derivative of the Energy).

Fix $\alpha>\beta>0$ and let $\mu\in\mathcal{P}_{c}(\mathbb{R}^{n})$ be a
fixed, compactly supported probability measure. Then

$2\frac{d^{2}}{d\beta^{2}}\mathcal{E}_{\alpha,\beta}[\mu]=-\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\frac{|x-y|^{\beta}((\log(|x-y|^{\beta}))^{2}-2\log(|x-y|^{\beta})+2)}{\beta^{3}}d\mu(x)d\mu(y),$

with the convention that $0((\log 0)^{2}-2\log(0)+2)=0.$

######  Proof 2.13.

This is more or less a direct application of the dominated convergence
theorem. To do so, we recall that, as we calculated in the proof of
Proposition 1.5,

$\frac{d}{d\beta}W_{\alpha,\beta}(x,y)=\begin{cases}0&\text{ if }x=y\\\
\frac{|x-y|^{\beta}(1-\log(|x-y|^{\beta}))}{\beta^{2}}&\text{ if }x\neq
y.\end{cases}$

and

$\frac{d^{2}}{d\beta^{2}}W_{\alpha,\beta}(x,y)=\begin{cases}0&\text{ if
}x=y\\\
-\frac{|x-y|^{\beta}((\log(|x-y|^{\beta}))^{2}-2\log(|x-y|^{\beta})+2)}{\beta^{3}}&\text{
if }x\neq y.\end{cases}.$

Each of these derivatives is continuous, and hence bounded on the compact set
$\operatorname{spt}(\mu\otimes\mu),$ which allows us to apply the Lebesgue
dominated convergence theorem as in [24, Theorem 2.27(b)] to recover the
desired result.

Applying this calculation, along with some small estimates, yields the
following differential inequality, which may be of interest in its own right:

######  Lemma 2.14 (Differential Inequality for the Energy).

Let $\alpha>\beta>0$ and let $\mu\in\mathcal{P}_{c}(\mathbb{R}^{n}).$ Then

$\frac{d^{2}}{d\beta^{2}}\mathcal{E}_{\alpha,\beta}[\mu]\leq\frac{1}{\beta^{2}}\mathcal{E}_{\alpha,\beta}[\mu].$

######  Proof 2.15.

By applying Lemma 2.12 and noticing that $s^{2}-2s+2\geq 1$ for all
$s\in\mathbb{R},$ we deduce that

$\displaystyle\frac{d^{2}}{d\beta^{2}}\mathcal{E}_{\alpha,\beta}[\mu]$
$\displaystyle\leq\frac{1}{2}\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}-\frac{|x-y|^{\beta}}{\beta^{3}}d\mu(x)d\mu(y)$
$\displaystyle\leq\frac{1}{2\beta^{2}}\iint_{\mathbb{R}^{n}\times\mathbb{R}^{n}}\frac{|x-y|^{\alpha}}{\alpha}-\frac{|x-y|^{\beta}}{\beta}d\mu(x)d\mu(y)$
$\displaystyle=\frac{1}{\beta^{2}}\mathcal{E}_{\alpha,\beta}(\mu).$

Finally, we can use the preceding differential inequality, along with a
comparison to the simplex, to deduce the following uniform bound.

######  Lemma 2.16 (Concavity Parameter of the Energy).

Let $0<\beta_{0}<\beta<\beta_{1}\leq\alpha$ and assume that
$\mu\in\mathcal{P}_{c}(\mathbb{R}^{n})$ satisfies
$\mathcal{E}_{\alpha,\beta}[\mu]\leq\frac{1}{2}\frac{n}{n+1}\left[\frac{1}{\alpha}-\frac{1}{\beta_{1}}\right].$
Then

$\frac{d^{2}}{d\beta^{2}}\mathcal{E}_{\alpha,\beta}[\mu]\leq\frac{1}{2\beta_{1}^{2}}\frac{n}{n+1}\left[\frac{1}{\alpha}-\frac{1}{\beta_{1}}\right].$

######  Proof 2.17.

By Lemma 2.14 and the assumption, we have that

$\frac{d^{2}}{d\beta^{2}}\mathcal{E}_{\alpha,\beta}[\mu]\leq\frac{1}{\beta^{2}}\mathcal{E}_{\alpha,\beta}[\mu]\leq\frac{1}{2\beta^{2}}\frac{n}{n+1}\left[\frac{1}{\alpha}-\frac{1}{\beta_{1}}\right].$

Since the rightmost term in this inequality is an increasing function of
$\beta$ on $[0,\alpha],$ we conclude that

$\frac{d^{2}}{d\beta^{2}}\mathcal{E}_{\alpha,\beta}[\mu]\leq\frac{1}{2\beta^{2}}\frac{n}{n+1}\left[\frac{1}{\alpha}-\frac{1}{\beta}\right]\leq\frac{1}{2\beta_{1}^{2}}\frac{n}{n+1}\left[\frac{1}{\alpha}-\frac{1}{\beta_{1}}\right],$

as desired.

We conclude by proving Proposition 1.7, which is based on the strong concavity
of $E_{\alpha}.$

######  Proof 2.18 (Proof of Proposition 1.7).

Define $\mathcal{F}:=\left\\{\mu\in\mathcal{P}_{c}(\mathbb{R}^{n})\ |\
\mathcal{E}_{\alpha,\beta}[\mu]\leq\frac{1}{2}\frac{n}{n+1}\left[\frac{1}{\alpha}-\frac{1}{\beta_{1}}\right]\right\\}.$
Note that $\mathcal{F}$ is non-empty because it contains a regular unit
simplex $\nu,$ as

$\mathcal{E}_{\alpha,\beta}[\nu]=\frac{1}{2}\frac{n}{n+1}\left[\frac{1}{\alpha}-\frac{1}{\beta}\right]\leq\frac{1}{2}\frac{n}{n+1}\left[\frac{1}{\alpha}-\frac{1}{\beta_{1}}\right].$

Thus, as an immediate consequence of the defintion of $\mathcal{F},$ we have
that $E^{\mathcal{F}}_{\alpha}(\beta)=E_{\alpha}(\beta)$ for all
$\beta\in[\beta_{0},\beta_{1}].$ In addition, by Lemma 2.16, we see that for
each $\mu\in\mathcal{F},$ $\mathcal{E}_{\alpha,\beta}[\mu]$ is strongly
concave, with concavity parameter
$\frac{1}{2}\frac{n}{n+1}\left[\frac{1}{\alpha}-\frac{1}{\beta_{1}}\right].$
Thus, by Lemma 2.10, we conclude that
$E_{\alpha}(\beta)=E_{\alpha}^{\mathcal{F}}(\beta)=\inf_{\mu\in\mathcal{F}}\mathcal{E}_{\alpha,\beta}[\mu]$
is strongly concave with the correct concavity parameter.

######  Remark 2.19.

We note that the results of Proposition 1.7 can be applied to any of the cases
described in (2.9), although if $\alpha=\beta_{1},$ then some more work is
required to obtain bounds which are better than those described in Lemma 2.5.
In particular, we can (1) fix $\beta_{1}^{\prime}\in(\beta_{0},\beta_{1})$ and
use 1.7 to estimate $E_{\alpha}(\beta_{1}^{\prime})$ from below and (2) use
(1.4) to estimate $E_{\alpha}(\beta_{t})$ in terms of $E_{\alpha}(\beta_{0})$
and $E_{\alpha}(\beta_{1}^{\prime}),$ then replace
$E_{\alpha}(\beta_{1}^{\prime})$ with our lower bound.

##  3 Cascading Phase Transitions

For the purposes of completeness, we prove the following technical lemma,
since we were unable to find a reference:

######  Lemma 3.1 (Compactness of $\mathcal{D}^{k}$).

Let $\Omega\subseteq\mathbb{R}^{n}$ be compact. Then for any $k\in\mathbb{N}$,
the set $\mathcal{D}^{k}(\Omega)$ is weakly compact (in duality with
$\mathcal{C}^{b}(\Omega)).$

######  Proof 3.2.

It is not difficult to check that, since $\Omega$ is compact,
$\mathcal{P}(\Omega)$ is also compact with respect to the weak topology (see,
for example, [25, Proposition 7.2.1]). Thus, we need only to check that
$\mathcal{D}^{k}(\Omega)$ is weakly closed. To this end, assume that we have a
sequence $\\{\mu_{m}\\}_{m}\subseteq\mathcal{D}^{k}(\Omega)$ such that
$\mu_{m}\rightharpoonup\mu_{\infty}\in\mathcal{P}(\Omega).$ Assume by way of
contradiction that $\mu_{\infty}\notin\mathcal{D}^{k}(\Omega),$ so that there
are $k+1$ distinct points $x_{1},\dots,x_{k+1}$ in
$\operatorname{spt}(\mu_{\infty}),$ and choose
$\varepsilon=\frac{1}{3}\min\\{|x_{i}-x_{j}|\ |\ i\neq j\\}.$ Moreover, for
$i\in\\{1,\dots,k+1\\},$ let $\eta_{i}$ be a (smooth and non-negative) bump
function such that $\eta_{i}(x_{i})=1$ and
$\operatorname{spt}(\eta_{i})\subseteq B_{\varepsilon}(x_{i}).$ For any given
$i\in\\{1,\dots,k+1\\},$ the assumption that
$x_{i}\in\operatorname{spt}(\mu_{\infty}),$ we have that $\int\eta_{i}\
d\mu_{\infty}>0.$ Moreover, since $\mu_{m}\rightharpoonup\mu_{\infty},$ we
must also have that, for sufficiently large $m,$ $\int\eta_{i}\ d\mu_{m}>0$
and hence that
$B_{\varepsilon}(x_{i})\cap\operatorname{spt}(\mu_{m})\neq\varnothing.$Thus,
by choosing $m$ large enough, we have that
$B_{\varepsilon}(x_{i})\cap\operatorname{spt}(\mu_{m})\neq\varnothing$ for all
$i\in\\{1,\dots,k+1\\}.$ As $\varepsilon$ was chosen to make the balls
$B_{\varepsilon}(x_{i})$ disjoint, we deduce that
$|\operatorname{spt}(\mu_{m})|\geq k$ for $m$ sufficiently large enough, a
contradiction to the assumption that
$\\{\mu_{m}\\}_{m}\subseteq\mathcal{D}^{k}(\Omega).$ Thus, we conclude that
$\mu_{\infty}\in\mathcal{D}^{k}(\Omega)$ and hence that
$\mathcal{D}^{k}(\Omega)$ is weakly compact.

This observation, along with the work on cardinality [8, Theorem 1.1], and the
identification of explicit minimizers in [17, Theorem 2.1(c)] and [19, Theorem
1], allows us to prove Theorem 1.9. In other words, we show that, for all
interesting values of $\alpha,$ the cardinality of minimizers of
$\mathcal{E}_{\alpha,\beta}$ tends to $\infty$ as $\beta\searrow 2.$

######  Proof 3.3 (Proof of Theorem 1.9).

We first treat the $n\geq 2$ case. As such, fix $\alpha\in(2,4)$ and by way of
contradiction, assume that the conclusion of the theorem does not hold. Then
there exists an integer $k\in\mathbb{N}$ and a sequence $\beta_{m}\searrow 2$
such that, for each $m,$
$E^{\mathcal{D}^{k}(\mathbb{R}^{n})}_{\alpha}(\beta_{m})=E_{\alpha}(\beta_{m})$
so that, by the continuity we proved in Proposition 1.5,

$E_{\alpha}^{\mathcal{D}^{k}(\mathbb{R}^{n})}(2)=\lim_{m\to\infty}E_{\alpha}^{\mathcal{D}^{k}(\mathbb{R}^{n})}(\beta_{m})=\lim_{m\to\infty}E_{\alpha}(\beta_{m})=E_{\alpha}(2).$

In particular, there exists a minimizing sequence
$\\{\mu_{i}\\}_{i}\subseteq\mathcal{D}^{k}(\mathbb{R}^{n})$ such that
$\lim_{i\to\infty}\mathcal{E}_{\alpha,2}(\mu_{i})=E_{\alpha}^{(}2).$ As [17,
Proposition 2.1] implies that all minimizers of $\mathcal{E}_{\alpha,2}$ have
support with diameter at most $e,$ and by the translation invariance of
$\mathcal{E}_{\alpha,2},$ we may assume that
$\\{\mu_{i}\\}_{i}\subseteq\mathcal{D}^{k}(\overline{B_{{e}}(0)}),$ which is
weakly compact by Lemma 3.1. Thus we can find that a subsequence of
$\\{\mu_{i}\\}_{i}$ weakly converges to some
$\mu_{\infty}\in\mathcal{D}^{k}(\overline{B_{{e}}(0)}).$ In particular, since
$\mathcal{E}_{\alpha,2}$ is weakly continuous (due, to, for example, [2,
Proposition 7.2]), we have that
$\mu_{\infty}\in\mathcal{D}^{k}(\overline{B_{{e}}(0)})\cap\operatorname{argmin}_{\mathcal{P}(\mathbb{R}^{n})}\mathcal{E}_{\alpha,2}[\cdot].$
However, this is a contradiction to [17, Theorem 2.1(c)], which asserts that
$\operatorname{argmin}_{\mathcal{P}(\mathbb{R}^{n})}\mathcal{E}_{\alpha,2}[\cdot]$
consists only of ‘spherical shell’ measures, whose supports have infinite
cardinality.

The proof in the case $n=1$ is nearly identical, with the only distinctions
being that we must choose $\alpha\in(2,3)$ and that we instead draw a
contradiction to [19, Theorem 1].

In light of the results in [8], where the authors show that for all
$\alpha>\beta>2,$ any minimizer $\mu$ of $\mathcal{E}_{\alpha,\beta}$ is
supported on a set of finite cardinality, Theorem 1.9 implies that, for any
fixed $\alpha$ in the range of validity of that theorem, there are an infinite
number of phase transitions as $\beta\searrow 2$, and the number of points in
the support of any minimizers of $\mathcal{E}_{\alpha,\beta}$ is forced to
grow to $\infty.$

With the goal of studying weak limits of minimizers of
$\mathcal{E}_{\alpha,\beta}$ for $\beta$ near $2,$ we also prove the following
technical lemma:

######  Lemma 3.4 (Minimizing Sequences for $\mathcal{E}_{\alpha,\beta}$).

Let $\alpha>\beta>0$ and let $\\{\beta_{m}\\}_{m}\subseteq(0,\alpha)$ with
$\beta_{m}\to\beta.$ Moreover, take
$\mu_{m}\in\operatorname{argmin}_{\mathcal{P}(\mathbb{R}^{n})}\mathcal{E}_{\alpha,\beta_{m}}[\cdot].$
Then $\mu_{m}$ is a minimizing sequence for $\mathcal{E}_{\alpha,\beta}.$

######  Proof 3.5.

Pass to the subsequences $\\{\beta_{m(\ell)}\\}_{\ell}$ and
$\\{\mu_{m(\ell)}\\}_{\ell}.$ By assumption,
$E_{\alpha}(\beta_{m(\ell)})=E_{\alpha}^{\\{\mu_{m(\ell)}\\}_{\ell}}(\beta_{m(\ell)})$
for any $\ell\in\mathbb{N},$ so by Proposition 1.5 and the fact that
$\beta_{m}\to\beta,$ we have that

$E_{\alpha}(\beta)=\lim_{\ell\to\infty}E_{\alpha}(\beta_{m(\ell)})=\lim_{\ell\to\infty}E_{\alpha}^{\\{\mu_{m(\ell)}\\}_{\ell}}(\beta_{m(\ell)})=E_{\alpha}^{\\{\mu_{m(\ell)}\\}_{\ell}}(\beta).$

In other words,
$\inf_{\ell\in\mathbb{N}}\mathcal{E}_{\alpha,\beta}[\mu_{m(\ell)}]=\inf_{\mu\in\mathcal{P}(\mathbb{R}^{n})}\mathcal{E}_{\alpha,\beta}[\mu_{m(\ell)}].$
Thus, there exists a further subsequence of $\\{\mu_{m(\ell)}\\}$ which is a
minimizing sequence for $\mathcal{E}_{\alpha,\beta},$ and hence $\mu_{m}$
itself is a minimizing sequence for $\mathcal{E}_{\alpha,\beta}.$

By taking the minimizing sequence in Theorem 1.9 to be of the form addressed
in Lemma 3.4, we deduce the following result which, in a relaxed sense which
will be made precise in the statement of the theorem, states that minimizers
of $\mathcal{E}_{\alpha,\beta},$ which are known to be discrete by [8],
converge weakly to spherical shells as $\beta\searrow 2:$

######  Theorem 3.6 (Weak Convergence to Spherical Shells).

Let $n\geq 2,$ and fix $\alpha\in(2,4).$ Let
$\\{\beta_{m}\\}_{m}\subseteq(2,\alpha)$ be such that $\beta_{m}\searrow 2,$
and define a sequence $\\{\mu_{m}\\}_{m}\subseteq\mathcal{P}(\mathbb{R}^{n})$
by
$\mu_{m}\in\operatorname{argmin}_{\mathcal{P}(\mathbb{R}^{n})}\mathcal{E}_{\alpha,\beta_{m}}.$
Then every subsequence of $\\{\mu_{m}\\}_{m}$ has a further subsequence which
weakly converges to some spherical shell.

######  Proof 3.7.

We first address the $n\geq 2$ case. Fix a subsequence
$\\{\mu_{m(\ell)}\\}_{\ell}$ of $\\{\mu_{m}\\}_{m}.$ As in the proof of
Theorem 1.9, we note that [17, Proposition 2.1] and translation invariance
allow us to assume that
$\\{\mu_{m(\ell)}\\}_{\ell}\subseteq\mathcal{P}(\overline{B_{e}(0)}).$ By
Lemma 3.4, $\\{\mu_{m(\ell)}\\}_{\ell}$ is a minimizing sequence for
$\mathcal{E}_{\alpha,2}.$ Hence, since $\mathcal{P}(\overline{B_{e}(0)})$ is
weakly compact and $\mathcal{E}_{\alpha,2}$ is weakly continuous, there exists
a further subsequence of $\\{\mu_{m(\ell)}\\}_{\ell}$ which converges to
$\mu_{\infty}\in\operatorname{argmin}_{\mathcal{P}(\mathbb{R}^{d})}\mathcal{E}_{\alpha,2}.$
Finally, by [17, Theorem 2.1(c)], $\mu_{\infty}$ is a spherical shell measure,
as desired.

The proof of the $n=1$ case is nearly identical, with the only difference
being that we use [19, Theorem 1] to conclude that $\mu_{\infty}$ is of the
form described in that theorem.

A related result holds in the one-dimensional case, where minimizers of
$\mathcal{E}_{\alpha,2}$ for $\alpha\in(2,3)$ are known by [19, Theorem 1]:

######  Proposition 3.8 (Weak Convergence in One Dimension).

Let $n=1$ and $\alpha\in(2,3).$ Let $\\{\beta_{m}\\}_{m}\in(1,\alpha)$ be such
that $\beta_{m}\to 2,$ and define a sequence
$\\{\mu_{m}\\}_{m}\subseteq\mathcal{P}(\mathbb{R}^{n})$ by
$\mu_{m}\in\operatorname{argmin}_{\mathcal{P}(\mathbb{R}^{n})}\mathcal{E}_{\alpha,\beta_{m}}.$
Then every subsequence of $\\{\mu_{m}\\}_{m}$ converges weakly to a measure
with density as described in [19, Theorem 1].

######  Proof 3.9.

For any subsequence of $\\{\mu_{m}\\}_{m},$ we use the same methods as in the
proof of Theorem 3.6 to recover a further subsequence which tends to
$\mu_{\infty}\in\operatorname{argmin}_{\mathcal{P}(\mathbb{R}^{d})}\mathcal{E}_{\alpha,2},$
and such minimizers are totally classified by [19, Theorem 1].

A second related result holds for $\alpha>4$ (if $n\geq 2$) and $\alpha>3$ (if
$n\geq 1$), which states that as $\beta\nearrow 2,$ minimizers of
$\mathcal{E}_{\alpha,\beta},$ whose supports necessarily have positive
Hausdorff dimension by [7, Theorem 1], converge to unit simplices studied in
[16] in the following sense:

######  Theorem 3.10 (Weak Convergence to Unit Simplices).

For $n=1$ fix $\alpha\geq 3$, and for $n\geq 2,$ fix $\alpha>4.$ Let
$\\{\beta_{m}\\}_{m}\subseteq(1,2)$ be such that $\beta_{m}\nearrow 2$ and
define a sequence $\\{\mu_{m}\\}_{m}\subseteq\mathcal{P}(\mathbb{R}^{n})$ by
$\mu_{m}\in\operatorname{argmin}_{\mathcal{P}(\mathbb{R}^{n})}\mathcal{E}_{\alpha,\beta_{m}}.$
Then every subsequence of $\\{\mu_{m}\\}_{m}$ has a further subsequence which
weakly converges to some regular unit simplex.

######  Proof 3.11.

By an argument identical to that the proof of Theorem 3.6, every subsequence
of $\\{\mu_{m}\\}_{m}$ has a further subsequence converging to
$\mu_{\infty}\in\operatorname{argmin}_{\mathcal{P}(\mathbb{R}^{d})}\mathcal{E}_{\alpha,2}.$
If $n\geq 2$ and $\alpha>4,$ then by [16, Corollary 1.4], $\mu_{\infty}$ is a
unit simplex, as desired. If, instead, $n=1$ and $\alpha\geq 3,$ then [17,
Corollary 2.3(c-d)] implies that $\mu_{\infty}$ is a unit simplex.

######  Remark 3.12.

We note that Theorem 3.6, Proposition 3.8, and Theorem 3.10 address all cases
where global minimizers of $\mathcal{E}_{\alpha,\beta}$ are not explicitly
known for $\alpha>\beta\approx 2.$ In particular, if $n\geq 2$,
$\alpha\in[2,4],$ and $\frac{-n^{2}+3n+2}{n+1}\leq\beta<2,$ then spherical
shells are the only global minimizers of $\mathcal{E}_{4,\beta}$ per [20,
Theorem 1]. If $(\alpha,\beta)=(4,2)$, then global minimizers of
$\mathcal{E}_{4,2}$ are precisely those probability measures which concentrate
on appropriate spherical shells and which satisfy the second-moment condition
of [16, Theorem 1]. Finally, $n\geq 2$ and $\alpha>4$, or if $n=1$ and
$\alpha\geq 3,$ then for $2<\beta<\alpha,$ regular unit simplex measures are
the only global minimizers of $\mathcal{E}_{4,\beta},$ this time by [16,
Theorem 1.2] and either [16, Theorem 1.1](in the $n\geq 2$ case) or [17,
Corollary 2.3(d)] (in the $n=1$ case).

##  4 Bound on Simplex Transition Threshold

We conclude by presenting the last contribution in our paper, which concerns
the transition threshold function $\alpha_{\Delta^{n}}(\beta)$ defined in [16,
Theorem 1.5]. Recall that, for $\beta\geq 2,$ $\alpha_{\Delta^{n}}(\beta)$ is
defined as the unique value of $\alpha$ such that $E_{\alpha,\beta}$ is
minimized by unit simplices for $\alpha>\alpha_{\Delta^{n}}(\beta),$ and unit
simplices do not minimize $E_{\alpha,\beta}$ for
$\alpha<\alpha_{\Delta^{n}}(\beta).$ While the value of
$\alpha_{\Delta^{n}}(\beta)$ is only explicitly known for $\beta=2,$ the work
in [16, Section 4.1] established an upper bound on
$\alpha_{\Delta^{n}}(\beta)$ which only depends on the dimension through
$\min(n,2)$ and [16, Section 4.2] established a dimensionally-dependent lower
bound. In the case where $n\geq 2,$ we provide an alternative lower bound,
which has two key advantages. First, the definition of the lower bound is more
easily explained than the bound in [16], which relies on a careful analysis
involving the Euler-Lagrange conditions. Second, we are able to show that our
lower bound is superior, at least in some cases, and have reason to believe
that this is the case in general. To begin, we recall the lower bound from
[16, Section 4].

######  Definition 4.1 (Lower Bounds and Unimodal Functions from [16]).

Let the function $f_{n}:(0,\infty)\to\mathbb{R}$ be defined by

$f_{n}(t):=\begin{cases}\frac{2^{-1}-2^{-t}}{t}&\text{ if }n=1\\\
\frac{n-(\frac{2n}{n+1})^{t/2}-n(\frac{n-1}{n+1})^{t/2}}{t}&\text{ if }n\geq
2.\end{cases}.$

For $\beta\geq 2,$ define $\underline{\alpha}_{\Delta^{n}}(\beta)$ by
$\underline{\alpha}_{\Delta^{n}}(\beta)=\max\\{\alpha\geq 2\ |\
\varphi_{n}(\alpha)=\varphi_{n}(\beta)\\}.$

Each $\underline{\alpha}_{\Delta^{n}}$ was proven to be a lower bound for the
threshold $\alpha_{\Delta}^{n}$ in [16, Proposition 4.12]. In particular,
those bounds were chosen by choosing points where we felt it likely for unit
simplices to violate the Euler-Lagrange conditions for a large range of
parameters, and then finding the range of parameters for which there is an
Euler-Lagrage violation. Our current approach, on the other hand, is based on
identifying a likely family of competitor measures, and checking for which
parameters these competitors have lower energy than the simplex. In light of
the translation and rotation invariance of this minimization problem, we
consider the cross-polytope measures of Definition 1.1. Of course, that
definition defines a one-parameter family of measures, so it is useful to find
the measure in this family with minimal energy:

######  Proposition 4.2 (Optimal Energy for Cross Polytopes).

Let $n\geq 1.$ Then $\mathcal{E}_{\alpha,\beta}[\mu^{*}_{n}(r)]$ is minimized
when

$r=r_{\alpha,\beta,n}^{*}=2^{-1/2}\left(\frac{2n-2+2^{\beta/2}}{2n-2+2^{\alpha/2}}\right)^{1/(\alpha-\beta)}.$

and

$\inf_{r}\mathcal{E}_{\alpha,\beta}[\mu^{*}_{n}(r)]=\frac{1}{4n}\left(\frac{(2n-2+2^{\beta/2})^{\alpha/(\alpha-\beta)}}{(2n-2+2^{\alpha/2})^{\beta/(\alpha-\beta)}}\right)\left(\frac{1}{\alpha}-\frac{1}{\beta}\right).$

######  Proof 4.3.

For any $r\geq 0,$ we compute

$\mathcal{E}_{\alpha,\beta}[\mu_{n}^{*}(r)]=\frac{1}{4n}\left[\left((2n-2)2^{\alpha/2}+2^{\alpha}\right)\frac{r^{\alpha}}{\alpha}-\left((2n-2)2^{\beta/2}+2^{\beta}\right)\frac{r^{\beta}}{\beta}\right],$
(4.1)

so that

$\frac{d}{dr}\mathcal{E}_{\alpha,\beta}[\mu_{n}^{*}(r)]=\frac{1}{4n}\left[\left((2n-2)2^{\alpha/2}+2^{\alpha}\right)r^{\alpha-1}-\left((2n-2)2^{\beta/2}+2^{\beta}\right)r^{\beta-1}\right].$

Solving, we see that
$\frac{d}{dr}\mathcal{E}_{\alpha,\beta}[\mu_{n}^{*}(r)]=0$ only for $r$
satisfying

$r^{\alpha-\beta}=\frac{(2n-2)2^{\beta/2}+2^{\beta}}{(2n-2)2^{\alpha/2}+2^{\alpha}}=2^{(\beta-\alpha)/2}\frac{2n-2+2^{\beta/2}}{2n-2+2{\alpha/2}},$

and solving for $r,$ we deduce that

$r=r^{*}_{\alpha,\beta,n}=2^{-1/2}\left(\frac{2n-2+2^{\beta/2}}{2n-2+2^{\alpha/2}}\right)^{1/(\alpha-\beta)}.$

Substituting this in to (4.1), after factoring out $r^{\beta}$, implies that

$\displaystyle\mathcal{E}_{\alpha,\beta}[\mu_{n}^{*}(r^{*}_{\alpha,\beta,n})]$
$\displaystyle=\frac{1}{4n}2^{-\beta/2}\left(\frac{2n-2+2^{\beta/2}}{2n-2+2^{\alpha/2}}\right)^{\beta/(\alpha-\beta)}((2n-2)2^{\beta/2}+2^{\beta})\left(\frac{1}{\alpha}-\frac{1}{\beta}\right)$
$\displaystyle=\frac{1}{4n}\left(\frac{(2n-2+2^{\beta/2})^{\alpha/(\alpha-\beta)}}{(2n-2+2^{\alpha/2})^{\beta/(\alpha-\beta)}}\right)\left(\frac{1}{\alpha}-\frac{1}{\beta}\right),$

as desired.

We will be interested in studying the relationship between
$\mathcal{E}_{\alpha,\beta}[\mu_{n}^{*}(r^{*}_{\alpha,\beta,n})]$ and
$\mathcal{E}_{\alpha,\beta}[\mu_{n}^{\Delta}].$ In particular, if
$\mathcal{E}_{\alpha,\beta}[\mu_{n}^{*}(r^{*}_{\alpha,\beta,n})]<\mathcal{E}_{\alpha,\beta}[\mu_{n}^{\Delta}],$
the optimal cross-polytope has lower interaction energy than any unit simplex,
so unit simplices cannot globally minimize $\mathcal{E}_{\alpha,\beta},$ so
that $\alpha_{\Delta^{n}}(\beta)\geq\alpha.$ To analyze when this occurs, we
will define a family of unimodal functions, analogously to the approach used
to find bounds in [16, Section 4].

######  Definition 4.4 (A New Family of Unimodal Functions).

For $n\geq 1$, define $\varphi_{n}:(0,\infty)\to(-\infty,0)$ by

$\varphi_{n}(\gamma):=-\left(\frac{(n+1)(2n-2+2^{\gamma/2})}{2n^{2}}\right)^{1/\gamma}.$

We note that, if $n=1,$ then $f_{1}\equiv-\sqrt{2}$ which, as will become
clear in the following discussion, is a consequence of the fact that the
optimal cross-polytope and the unit simplex coincide in the one-dimensional
case. We now state a couple of relevant properties of the $\varphi_{n}.$

######  Lemma 4.5 (Unimodality of $\mathcal{\varphi}_{n}$).

Let $n\geq 2.$ Then $\varphi_{n}$ is a unimodal function, in the sense that it
has a unique global maximum and no other critical points. Moreover, if
$t_{0}\in(1+\frac{n-1}{2n^{2}},\infty)$ is defined as the unique solution of

$\left(\frac{2n^{2}}{n+1}t_{0}-(2n-2)\right)\log\left(\frac{2n^{2}}{n+1}t_{0}-(2n-2)\right)=\frac{2n^{2}}{n+1}t_{0}\log
t_{0}$

on $(1+\frac{n-1}{2n^{2}},\infty)$, then $\varphi_{n}$ is maximized at
$\gamma_{0}=\frac{2\log\left(\frac{2n^{2}}{n+1}t_{0}-(2n-2)\right)}{\log 2}.$

######  Proof 4.6.

Postponed to the appendix.

######  Lemma 4.7 ($\varphi_{n}$ Characterizes Competition between Simplices
and Cross Polytopes).

Let $\alpha>\beta>0$, and let $n\geq 1.$ Then
$\varphi_{n}(\alpha)-\varphi_{n}(\beta)$ and
$\mathcal{E}_{\alpha,\beta}[\mu_{n}^{*}(r_{\alpha,\beta,n}^{*})]-\mathcal{E}_{\alpha,\beta}[\mu_{n}^{\Delta}]$
have opposite sign.

######  Proof 4.8.

Applying Proposition 4.2 and using that the energy of the unit simplex is
$\frac{1}{2}\frac{n}{n+1}\left(\frac{1}{\alpha}-\frac{1}{\beta}\right),$ we
see that

$\mathcal{E}_{\alpha,\beta}[\mu_{n}^{*}(r_{\alpha,\beta,n}^{*})]-\mathcal{E}_{\alpha,\beta}[\mu_{n}^{\Delta}]=\frac{1}{4n}\left(\frac{(2n-2+2^{\beta/2})^{\alpha/(\alpha-\beta)}}{(2n-2+2^{\alpha/2})^{\beta/(\alpha-\beta)}}\right)\left(\frac{1}{\alpha}-\frac{1}{\beta}\right)-\frac{1}{2}\frac{n}{n+1}\left(\frac{1}{\alpha}-\frac{1}{\beta}\right).$

Multiplying through by the positive quantity
$4n\left(\frac{1}{\beta}-\frac{1}{\alpha}\right)^{-1},$ we see that this
quantity has the same sign as

$\frac{2n^{2}}{n+1}-\frac{(2n-2+2^{\beta/2})^{\alpha/(\alpha-\beta)}}{(2n-2+2^{\alpha/2})^{\beta/(\alpha-\beta)}}.$

Since the map $t\mapsto t^{\alpha-\beta}$ is injective and increasing on
$(0,\infty),$ we may apply this map to both terms in the preceding line to
find that
$\mathcal{E}_{\alpha,\beta}[\mu_{n}^{*}(r_{\alpha,\beta,n}^{*})]-\mathcal{E}_{\alpha,\beta}[\mu_{n}^{\Delta}]$
has the same sign as

$\left(\frac{2n^{2}}{n+1}\right)^{\alpha-\beta}-\frac{(2n-2+2^{\beta/2})^{\alpha}}{(2n-2+2^{\alpha/2})^{\beta}}.$

Multiplying through by
$(2n-2+2^{\alpha/2})^{\beta}\left(\frac{n+1}{2n^{2}}\right)^{\alpha}$ and
raising both sides to the power of $\frac{1}{\alpha\beta},$ we conclude that
$\mathcal{E}_{\alpha,\beta}[\mu_{n}^{*}(r_{\alpha,\beta,n}^{*})]-\mathcal{E}_{\alpha,\beta}[\mu_{n}^{\Delta}]$
has the same sign as

$\left(\frac{(n+1)(2n-2+2^{\alpha/2})}{2n^{2}}\right)^{1/\alpha}-\left(\frac{(n+1)(2n-2+2^{\beta/2})}{2n^{2}}\right)^{1/\beta}=-(\varphi_{n}(\alpha)-\varphi_{n}(\beta)).$

######  Proof 4.9 (Proof of Proposition 1.10).

Let $\alpha\in(\beta,\underline{\alpha}_{\Delta^{n}}^{*}(\beta)).$ By
unimodality of $\varphi_{n},$ $\varphi_{n}(\alpha)-\varphi_{n}(\beta)>0.$
Thus, by Lemma 4.7,
$\mathcal{E}_{\alpha,\beta}[\mu_{n}^{*}(r_{\alpha,\beta,n}^{*})]-\mathcal{E}_{\alpha,\beta}[\mu_{n}^{\Delta}]<0,$
so that the optimal cross-polytope has strictly lower energy than the unit
simplex. Thus, by the definition of $\alpha_{\Delta^{n}}$ in [16, Theorem
1.5], $\alpha<\alpha_{\Delta^{n}}(\beta),$ as desired.

We now turn our attention to providing some non-rigorous evidence which
suggests that $\underline{\alpha}_{\Delta^{n}}^{*}$ is a better family of
bounds than $\underline{\alpha}_{\Delta^{n}}.$

######  Example 4.10.

Let $n=2$ and $\beta=2.5.$ Then we can check numerically that
$\underline{\alpha}^{*}_{\Delta^{2}}(2.5)\approx 3.18$, but
$\underline{\alpha}_{\Delta^{2}}(2.5)\approx 3.07.$

Figure 1: A comparison of $\underline{\alpha}_{\Delta^{2}}^{*}$ and
$\underline{\alpha}_{\Delta^{2}}$ appears to indicate that
$\underline{\alpha}_{\Delta^{2}}^{*}\geq\underline{\alpha}_{\Delta^{2}}$ for
all $\beta>2$ with $\alpha_{\Delta^{2}}^{*}(\beta)>\beta$. This improvement
allows us to expand the area in parameter space where unit simplices are known
not to minimize $\mathcal{E}_{\alpha,\beta},$ which is represented in this
figure as the red region to the left of the graph of
$\underline{\alpha}_{\Delta^{n}}.$ Numerical experiments for higher dimensions
$(n\geq 3)$ also showed similar results, which leads us to believe that, for
any $n\geq 2,$
$\underline{\alpha}_{\Delta^{n}}^{*}\geq\underline{\alpha}_{\Delta^{n}}$ for
all $\beta>2$ with $\alpha_{\Delta^{n}}^{*}(\beta)>\beta.$

These pieces of evidence lead us to the following two conjectures, which we
have not been able to prove. We begin with the weaker conjecture, which is
also well-supported by numerical evidence first.

######  Conjecture 4.11 (Superiority of
$\underline{\alpha}_{\Delta^{n}}^{*}$).

Let $n\geq 2.$ Then
$\underline{\alpha}_{\Delta^{n}}^{*}(\beta)\geq\underline{\alpha}_{\Delta^{n}}(\beta)$
for any $\beta\geq 2,$ with equality if and only if $\beta=2$ or
$\alpha_{\Delta^{n}}^{*}(\beta)=\beta.$

We conclude with the following stronger conjecture, which states that
$\underline{\alpha}_{\Delta^{n}}^{*}$ is actually the threshold function:

######  Conjecture 4.12 (Sharpnessof $\underline{\alpha}_{\Delta^{n}}^{*}$).

Let $n\geq 2$ and $\beta\geq 2.$ Then
$\underline{\alpha}_{\Delta^{n}}^{*}(\beta)=\alpha_{\Delta^{n}}(\beta).$

##  Appendix A Unimodality of $\varphi_{n}$

We now include the postponed proof of Lemma 4.5. To prove this lemma, it will
be useful to recall the following properties of unimodal functions:

######  Lemma A.1 (Relevant Properties of Unimodal Functions).

Let $g:(a,b)\to(c,d)$ and $h:(c,d)\to\mathbb{R}$ be $C^{1}.$

  * •

If $g$ is strictly increasing, then $h$ is unimodal on $(c,d)$ if and only if
$h\circ g$ is unimodal on $(a,b).$ In this case, if $h\circ g$ is maximized at
$t_{0}\in(a,b),$ then $h$ is maximized at $g(t_{0})\in(c,d).$

  * •

If $h$ is strictly increasing on $(c,d),$ then $g$ is unimodal on $(a,b)$ if
and only if $h\circ g$ is unimodal on $(a,b).$ In this case, $g$ and $h\circ
g$ are both maximized at the same point $t_{0}.$

######  Proof A.2.

Both these facts are immediate consequences of the chain rule.

######  Proof A.3 (Proof of Lemma 4.5).

Define a strictly increasing function $\ell:(-\infty,0)\to(-\infty,\infty)$ by
$\ell(s)=-\frac{2\log(-x)}{\log 2}$ and, for each $n\geq 2,$ a strictly
increasing function
$\xi_{n}:\left(1+\frac{n-1}{2n^{2}},\infty\right)\to(0,\infty)$ by

$\xi_{n}(t)=\frac{2\log\left(\frac{2n^{2}}{n+1}t-(2n-2)\right)}{\log 2}.$

By applying Lemma A.1 twice, $\varphi_{n}$ is unimodal on $(0,\infty)$ if and
only if $\ell\circ\varphi_{n}\circ\xi_{n}$ is unimodal on
$(1+\frac{n-1}{2n^{2}},\infty).$ After calculating, we see that

$\ell\circ\varphi_{n}\circ\xi_{n}(t)=-\frac{\log
t}{\log\left(\frac{2n^{2}}{n+1}t-(2n-2)\right)},$

so we will only need to check that this function is unimodal on
$(1+\frac{n-1}{2n^{2}},\infty)$ in order to prove the lemma.

Now, to check unimodality, we compute
$(\ell\circ\varphi_{n}\circ\xi_{n})^{\prime}(t)$. In particular, we have that

$(\ell\circ\varphi_{n}\circ\xi_{n})^{\prime}(t)=-\frac{t^{-1}\log\left(\frac{2n^{2}}{n+1}t-(2n-2)\right)-\frac{2n^{2}}{n+1}\left(\frac{2n^{2}}{n+1}t-(2n-2)\right)^{-1}\log
t}{\log^{2}\left(\frac{2n^{2}}{n+1}t-(2n-2)\right)}.$

Multiplying both sides by the positive quantity
$t\left(\frac{2n^{2}}{n+1}t-(2n-2)\right)\log^{2}\left(\frac{2n^{2}}{n+1}t-(2n-2)\right),$
we find that $(\ell\circ\varphi_{n}\circ\xi_{n})^{\prime}(t)$ has the same
sign as

$g_{n}(t):=-\left(\frac{2n^{2}}{n+1}t-(2n-2)\right)\log\left(\frac{2n^{2}}{n+1}t-(2n-2)\right)+\frac{2n^{2}}{n+1}t\log
t$

on $(1+\frac{n-1}{2n^{2}},\infty).$

As such, to conclude unimodality, it will suffice to show that $g_{n}$ is
concave on $(1+\frac{n-1}{2n^{2}},\infty)$ with
$g_{n}(1+\frac{n-1}{2n^{2}})>0.$ For concavity, we compute derivatives. That
is,

$\displaystyle g_{n}^{\prime}(t)$
$\displaystyle=-\frac{2n^{2}}{n+1}\log\left(\frac{2n^{2}}{n+1}t-(2n-2)\right)-\frac{2n^{2}}{n+1}\frac{\frac{2n^{2}}{n+1}t-(2n-2)}{\frac{2n^{2}}{n+1}t-(2n-2)}+\frac{2n^{2}}{n+1}\log
t+\frac{2n^{2}}{n+1}$
$\displaystyle=-\frac{2n^{2}}{n+1}\log\left(\frac{2n^{2}}{n+1}t-(2n-2)\right)+\frac{2n^{2}}{n+1}\log
t.$

and

$\displaystyle g_{n}^{\prime\prime}(t)$
$\displaystyle=-\left(\frac{2n^{2}}{n+1}\right)^{2}\left(\frac{2n^{2}}{n+1}t-(2n-2)\right)^{-1}+\frac{2n^{2}}{n+1}t^{-1}$
$\displaystyle=-\frac{\frac{2n^{2}}{n+1}(2n-2)}{t\left(\frac{2n^{2}}{n+1}t-(2n-2)\right)},$

which allows us to readily note that $g_{n}$ is concave on the desired
interval, as $\frac{2n^{2}}{n+1}$, $2n-2,$ $t$, and
$\frac{2n^{2}}{n+1}t-(2n-2)$ are all positive due to our choice of interval
and $n$. Finally, we compute

$\displaystyle g_{n}\left(1+\frac{n-1}{2n^{2}}\right)$
$\displaystyle=(2n-1)\log\left(1+\frac{n-1}{2n^{2}}\right),$

which is positive since $2n-1$ and $\frac{n-1}{2n^{2}}$ are both positive.

To see where $\varphi_{n}$ is maximized, we use basic calculus to note that
the unique maximum $t_{0}$ of $\ell\circ\varphi_{n}\circ\xi_{n}$ on
$(1+\frac{n-1}{2n^{2}},\infty)$ is defined by the relation $g_{n}(t)=0,$ or
rather,

$\left(\frac{2n^{2}}{n+1}t_{0}-(2n-2)\right)\log\left(\frac{2n^{2}}{n+1}t_{0}-(2n-2)\right)=\frac{2n^{2}}{n+1}t_{0}\log
t_{0}.$

Applying Lemma A.1 twice, we see that $\varphi_{n}$ is uniquely maximized at

$\xi_{n}(t_{0})=\frac{2\log\left(\frac{2n^{2}}{n+1}t_{0}-(2n-2)\right)}{\log
2}.$

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