Absolute Minima of Potentials
of a Certain Class of Spherical Designs

Sergiy Borodachov

Department of Mathematics, Towson University, Towson, MD, 21252

###### Abstract

We use linear programming techniques to find points of absolute minimum over
the unit sphere $S^{d}$ in $\mathbb{R}^{d+1}$ of the total potential of a
point configuration $\omega_{N}\subset S^{d}$ which is a spherical
$(2m-1)$-design contained in the union of some $m$ parallel hyperplanes. The
interaction between points is described by the kernel $K({\bf x},{\bf
y})=f(\left|{\bf x}-{\bf y}\right|^{2})$, where $\left|\ \\!\cdot\ \\!\right|$
is the Euclidean norm in $\mathbb{R}^{d+1}$. The potential function $f$ is
assumed to have a convex derivative $f^{(2m-2)}$. Points of minimum do not
depend on $f$ and are those and only those which form exactly $m$ distinct dot
products with points of $\omega_{N}$. The proof of this theorem was presented
at a workshop at ESI in January 2022. Using this result, we find sets of
universal minima of certain six configurations on higher-dimensional spheres.

Keywords: Spherical design, sharp configuration, extrema of a potential,
universal extrema, Gegenbauer polynomials, absolutely (completely) monotone
potential, Hermite interpolation.

MSC 2020: 33C45, 41A05, 52B11, 52C99, 33D90, 31B99.

##  1 Introduction and setting of the problem

Let
$S^{d}:=\\{(x_{1},\ldots,x_{d+1})\in\mathbb{R}^{d+1}:x_{1}^{2}+\ldots+x_{d+1}^{2}=1\\}$
denote the unit sphere in $\mathbb{R}^{d+1}$. The problem of finding absolute
minimum points for the total potential over $S^{d}$ of a point configuration
(regular in some sense) was earlier studied in a number of works. In
particular, papers [28, 29, 25, 26] focused on a regular $N$-gon on $S^{1}$
and a regular simplex, regular cross-polytope, and cube on $S^{d}$. Here we
extend the results of these papers for the absolute minimum to a class of
spherical designs which appears to contain quite a large number of other known
configurations. We use a general approach to characterizing the absolute
minima of the potential by first finding the Hermite interpolating polynomial
for the potential function and then applying properties of Gegenbauer
polynomials. This is the classical Delsarte-Yudin (or linear programming)
method, see the work by Cohn and Kumar [14], book [9], and references therein.

One important application of this problem is the maximal polarization problem
on the sphere, which requires maximizing the absolute minimum over $S^{d}$ of
the total potential of $N$ points lying on $S^{d}$. When the solution to the
maximal polarization problem is expected to be a regular configuration,
knowing the minimum value of its potential over the sphere would provide a
sharp upper bound to prove. Reviews of known results on polarization can be
found, for example, in book [9, Chapter 14] with most recent work reviewed in
[4].

Let $f:[0,4]\to(-\infty,\infty]$ be a function finite and continuous on the
interval $(0,4]$ with $\lim\limits_{t\to 0^{+}}f(t)=f(0)$. We call $f$ a
potential function. Later, we will specify additional assumptions on its
derivative. Points ${\bf x}$ and ${\bf y}$ on $S^{d}$ interact via the kernel
of the form $K({\bf x},{\bf y})=f(\left|{\bf x}-{\bf y}\right|^{2})$. Given a
point configuration $\omega_{N}=\\{{\bf x}_{1},\ldots,{\bf x}_{N}\\}\subset
S^{d}$, we define the $f$-potential of $\omega_{N}$ as

$p_{f}({\bf x},\omega_{N}):=\sum\limits_{i=1}^{N}f(\left|{\bf x}-{\bf
x}_{i}\right|^{2}),\ \ \ {\bf x}\in S^{d},$ (1)

and study the following problem.

######  Problem 1.1.

Find the quantity

$P_{f}(\omega_{N};S^{d}):=\min\limits_{{\bf x}\in S^{d}}p_{f}({\bf
x},\omega_{N}),$ (2)

and determine all points ${\bf x}^{\ast}\in S^{d}$, for which the minimum is
attained in (2).

Let $\omega_{N}^{\ast}$ be the configuration of the vertices of a regular
$N$-gon inscribed in $S^{1}$. We will call dual for $\omega_{N}^{\ast}$ the
regular $N$-gon formed by the midpoints of the arcs that join any two
consecutive vertices of $\omega_{N}^{\ast}$. Let $\omega_{d+1}^{\ast}$ be the
set of vertices of a regular simplex inscribed in $S^{d-1}$. The configuration
$-\omega_{d+1}^{\ast}$ will be called its dual. Let
$\omega_{2d}^{\ast}:=\\{\pm{\bf e}_{1},\ldots,\pm{\bf e}_{d}\\}$, $d\geq 3$,
where $\\{{\bf e}_{1},\ldots,{\bf e}_{d}\\}$ is the standard basis in
$\mathbb{R}^{d}$, be the set of vertices of the regular cross-polytope
inscribed in $S^{d-1}$, and $U_{d}$ be the set
$\left\\{\left(\pm\frac{1}{\sqrt{d}},\ldots,\pm\frac{1}{\sqrt{d}}\right)\right\\}\subset\mathbb{R}^{d}$
of $2^{d}$ vertices of the cube inscribed in $S^{d-1}$. The configurations
$\omega_{2d}^{\ast}$ and $U_{d}$ will be called dual for each other. Recall
that $\omega^{\ast}_{N}$ is a spherical $(N-1)$-design, $\omega_{d+1}^{\ast}$
is a $2$-design, while $\omega_{2d}^{\ast}$ and $U_{d}$ are both $3$-designs.

Early work. Stolarsky [28, 29] found the absolute minima and maxima for the
Riesz potential function $f(t)=-t^{-s/2}$ for a certain range of $s<0$ and the
vertices of a regular polygon, regular simplex, regular cross-polytope, and a
cube as well as absolute minimum for $f(t)=t^{-s/2}$, $s>0$, and the vertices
of a regular cross-polytope and a cube. Later, Nikolov and Rafailov [25, 26]
extended these results to all values of $s\neq 0$ for all four configurations
$\omega^{\ast}_{N}$, $\omega^{\ast}_{d+1}$, $\omega^{\ast}_{2d}$, and $U_{d}$
and potential function

$f(t)=\begin{cases}\ \ (t+C)^{-s/2},&s>0,\cr-(t+C)^{-s/2},&s<0,\cr\end{cases}\
\ \ C\geq 0.$ (3)

One of the following three cases was proved to be always true: the potential
of the configuration achieves its absolute minimum over $S^{d}$ at the points
of the dual configuration and its absolute maximum at the points of the
configuration itself, or vice versa, or the potential of the configuration is
constant over $S^{d}$. Which case holds depends on whether the derivative of
order one more than the spherical design strengh of the configuration is
positive, negative, or zero on $(0,4]$.

The paper is structured as follows. Section 2 contains known facts and
definitions used further in the paper while Section 3 reviews known results on
universal minima. In Section 4, we state the general results, Theorems 4.3 and
4.5, and in Section 5, we give their immediate consequences. We prove Theorems
4.3 and 4.5 in Section 6. Section 7 contains a known auxiliary statement on
the distribution of dot products that a universal minimum point forms with
points of a given stiff configuration. In Sections 8, 9, and 10, we find
universal minima of the three pairs of mutually dual stiff configurations
considered in this paper.

##  2 Preliminaries

In this section, we recall relevant definitions and known facts. An infinitely
differentiable function $f:(a,b)\to\mathbb{R}$ is called completely monotone
on $(a,b)$ if $(-1)^{k}f^{(k)}$ is non-negative on $(a,b)$ for all $k\geq 0$
and strictly completely monotone if $(-1)^{k}f^{(k)}$ is strictly positive on
$(a,b)$ for all $k\geq 0$. Such are potential functions (3) for $s>0$ and
$\widetilde{f}(t)=e^{-at}$ for $a>0$. Function (3) for $C=0$ and $s>0$ defines
the Riesz $s$-kernel $K_{s}({\bf x},{\bf y}):=\left|{\bf x}-{\bf
y}\right|^{-s}$ and function $\widetilde{f}$ the Gaussian kernel $K({\bf
x},{\bf y})=e^{-a\left|{\bf x}-{\bf y}\right|^{2}}$. Furthermore, potential
functions (3) for $-2<s<0$ and $f_{\log}(t)=\frac{1}{2}\ln\frac{1}{t}$ become
strictly completely monotone on $(0,4)$ after adding an appropriate positive
constant. Function (3) for $C=0$ and $s<0$ gives a rise to the Riesz kernel
$K_{s}({\bf x},{\bf y}):=-\left|{\bf x}-{\bf y}\right|^{-s}$ and function
$f_{\log}$ to the logarithmic kernel $K_{\log}({\bf x},{\bf
y}):=\ln\frac{1}{\left|{\bf x}-{\bf y}\right|}$.

Completely monotone $f$ on $(0,4)$ (modulo an additive constant) have convex
derivatives of even orders and concave derivatives of odd orders. Extended to
$[0,4]$ by limits at the endpoints, they satisfy the assumptions of Theorems
4.3 and 4.5.

Denote

$w_{d}(t):=\gamma_{d}(1-t^{2})^{d/2-1},$

where the constant $\gamma_{d}$ is chosen so that $w_{d}$ is a probability
density on $[-1,1]$. We recall that the Gegenbauer (or ultraspherical)
orthogonal polynomials corresponding to the sphere $S^{d}$ in
$\mathbb{R}^{d+1}$ are terms of the sequence
$\\{P_{n}^{(d)}\\}_{n=0}^{\infty}$ of univariate polynomials such that ${\rm
deg}\ \\!P_{n}^{(d)}=n$, $n\geq 0$, and

$\int_{-1}^{1}P_{i}^{(d)}(t)P_{j}^{(d)}(t)w_{d}(t)\ \\!dt=0,\ \ \ i\neq j,$

normalized so that $P_{n}^{(d)}(1)=1$, $n\geq 0$ (see [30, Chapter 4] or [9,
Chapter 5]). When $d=1$ or $d=2$, we obtain the classical Chebyshev
polynomials of the first kind or the Legendre polynomials, respectively. When
$d=3$, we have the Chebyshev polynomials of the second kind. It is a well-
known fact that $P_{n}^{(d)}$ has exactly $n$ distinct zeros in $(-1,1)$ (see
[30, Theorem 3.3.1]), which we will denote by

$-1<\kappa_{1}^{n}<\ldots<\kappa_{n}^{n}<1.$ (4)

Since $P_{n}^{(d)}$ is even for even $n$ and odd for odd $n$, its set of roots
is symmetric about $0$.

Let $\sigma_{d}$ denote the $d$-dimensional area measure on $S^{d}$ normalized
to be a probability measure; that is,
$\sigma_{d}\left(\cdot\right)=\frac{1}{\mathcal{H}_{d}(S^{d})}\mathcal{H}_{d}|_{S^{d}}\left(\cdot\right)$,
where $\mathcal{H}_{d}$ is the $d$-dimensional Hausdorff measure in
$\mathbb{R}^{d+1}$. Following the work by Delsarte, Goethals, and Seidel [16],
we call a point configuration $\omega_{N}$ a spherical $n$-design if, for
every polynomial $q$ on $\mathbb{R}^{d+1}$ of degree at most $n$,

$\frac{1}{N}\sum\limits_{i=1}^{N}q({\bf x}_{i})=\int_{S^{d}}q({\bf x})\
\\!d\sigma_{d}({\bf x}).$ (5)

The number $n$ is called the strength of the spherical design $\omega_{N}$.

Let $\mathbb{P}_{n}$ denote the space of all univariate polynomials of degree
$\leq n$. An equivalent definition of a spherical $n$-design, $n\geq 1$, on
$S^{d}$ is the fact [16] that the equality

$\sum\limits_{i=1}^{N}p({\bf x}\cdot{\bf x}_{i})=a_{0}(p)N,\ \ \ {\bf x}\in
S^{d},$ (6)

holds for every polynomial $p\in\mathbb{P}_{n}$. Here, $a_{0}(p)$ is the
$0$-th Gegenbauer coefficient of $p$.

Let $m\in\mathbb{N}$, ${\bf z}\in S^{d}$, and $-1\leq t_{1}<\ldots<t_{m}\leq
1$ be the distinct values of the dot product that ${\bf z}$ forms with points
of the $n$-design $\omega_{N}$. Denote by $M_{j}$ the number of points in
$\omega_{N}$ that form dot product $t_{j}$ with ${\bf z}$, $j=1,\ldots,m$.
Substituting ${\bf z}$ for ${\bf x}$ in (6) we obtain a quadrature with nodes
$t_{j}$ and weights $\beta_{j}:=M_{j}/N$ satisfying

$a_{0}(p)=\int_{-1}^{1}p(t)w_{d}(t)\
\\!dt=\sum\limits_{j=1}^{m}\beta_{j}p(t_{j}),\ \ \ p\in\mathbb{P}_{n}.$ (7)

Denote by $\\{\varphi_{1},\ldots,\varphi_{m}\\}$ the fundamental set of
polynomials for nodes $t_{1},\ldots,t_{m}$; that is,
$\varphi_{k}\in\mathbb{P}_{m-1}$, $\varphi_{k}(t_{k})=1$, and
$\varphi_{k}(t_{i})=0$, $i\neq k$, $k=1,\ldots,m$.

When $n\geq m-1$, the weights $\beta_{j}$ in (7) are uniquely defined (see,
e.g., [20]): $\beta_{j}=a_{0}(\varphi_{j})$, $j=1,\ldots,m$. To see this,
simply let $p=\varphi_{j}$ in (7). Such a quadrature (7) is called
interpolatory. Thus, $M_{j}=a_{0}(\varphi_{j})N$, $j=1,\ldots,m$; i.e.,
frequencies $M_{j}$ are determined by dot products $t_{1},\ldots,t_{m}$ and
the cardinality $N$ and do not depend on ${\bf z}$. Frequencies $M_{j}$ can
also be found as solutions to a certain Vandermonde linear system, see [12,
Theorem 2.1].

We next recall a well-known classification of quadratures with the highest
algebraic degree of precision and dot product distributions for the
corresponding classes of spherical designs (cf. [16]). These quadratures are
special cases of Gauss-Jacobi mechanical quadrature, see [30, Theorem 3.4.1].
The strict positivity of weights $\beta_{j}$ (called Christoffel numbers) in
quadratures described below follows from [30, Theorem 3.4.2].

Case I: $n=2m-1$. We will call $\omega_{N}$ $m$-stiff in this case, see
Definition 4.1. Quadrature (7) is now exact on $\mathbb{P}_{2m-1}$. The set
$\\{t_{1},\ldots,t_{m}\\}$ is the set of zeros of $P_{m}^{(d)}$. To see this,
simply let $p(t)=\varphi_{k}(t)P_{m}^{(d)}(t)$ and use the equality in (7).
Quadrature (7) is called the Gauss-Gegenbauer quadrature.

######  Remark 2.1.

From the above mentioned well-known facts, we immediately have the following.
For every point ${\bf z}\in S^{d}$ that forms $m$ distinct dot products with
points of a given $m$-stiff configuration, the values of dot products are
zeros of the $m$-th Gegenbauer polynomial $P_{m}^{(d)}$ and their frequencies
are independent of ${\bf z}$ and can be computed as
$M_{j}=a_{0}(\varphi_{j})N$.

Case II: $n=2m-2$, $m\geq 2$, and ${\bf z}\in\omega_{N}$. Then quadrature (7)
is exact on $\mathbb{P}_{2m-2}$ with $t_{m}=1$ and $\beta_{m}=1/N$. The nodes
$t_{1},\ldots,t_{m-1}$ are zeros of the Jacobi orthogonal polynomial
$J^{(\alpha,\beta)}_{m-1}$ of degree $m-1$, where $\alpha=d/2$ and
$\beta=d/2-1$. To see this, let
$p(t)=\varphi_{k}(t)J_{m-1}^{(\alpha,\beta)}(t)$, $k=1,\ldots,m-1$, in (7)
(which becomes $0$) and move the factor $1-t$ from $\varphi_{k}$ to the weight
$w_{d}$ in the integral.

Case III: $n=2m-3$, $m\geq 2$, ${\bf z}\in\omega_{N}$, and $\omega_{N}$ is
antipodal. Then quadrature (7) is exact on $\mathbb{P}_{2m-3}$ with
$t_{1}=-1$, $t_{m}=1$, and $\beta_{1}=\beta_{m}=1/N$. For $m\geq 3$, the nodes
$t_{2},\ldots,t_{m-1}$ are zeros of the Gegenbauer orthogonal polynomial
$P^{(d+2)}_{m-2}$. To see this, let $p(t)=\varphi_{k}(t)P_{m-2}^{(d+2)}(t)$,
$k=2,\ldots,m-1$, in (7) and move factors $1-t$ and $1+t$ from $\varphi_{k}$
to the weight in the integral.

######  Remark 2.2.

If $\omega_{N}$ is a strongly $m$-sharp or $m$-sharp antipodal configuration
on $S^{d}$ (see Definition 4.4), then each point in $\omega_{N}$ forms the
same set of $m$ dot products with other points of $\omega_{N}$ with the same
frequencies (which can be computed as described above). This set of dot
products is the set of zeros of the Jacobi polynomial
$J_{m}^{(\alpha,\beta)}$, where $\alpha=d/2$ and $\beta=d/2-1$, when
$\omega_{N}$ is strongly $m$-sharp and is the set of zeros of the Gegenbauer
polynomial $P_{m-1}^{(d+2)}$ together with $-1$ when $\omega_{N}$ is $m$-sharp
antipodal.

##  3 Review of known results on universal extrema

A number of recent works were devoted to finding extrema of general potentials
of regular configurations. Earlier results for power-law potentials were
reviewed in Section 1. Among kernels of the form $K({\bf x},{\bf
y})=f(\left|{\bf x}-{\bf y}\right|^{2})$, of greatest interest here are the
ones with a completely monotone potential function $f$ on $(0,4)$ or with $f$
that becomes completely monotone after adding a constant.

######  Definition 3.1.

Given a configuration $\omega_{N}\subset S^{d}$, a point ${\bf x}^{\ast}\in
S^{d}$ is called a universal minimum point (of the potential) of $\omega_{N}$
if ${\bf x}^{\ast}$ attains the absolute minimum in (2) for every potential
function $f$ completely monotone on $(0,4]$.

For universal minimality results obtained here, we will require only that the
derivative of $f$ of a certain even order be convex or the derivative of a
certain odd order be concave. Hardin, Kendall, and Saff [19] proved that
absolute minima of the potential of $\omega_{N}^{\ast}$ on $S^{1}$ with
respect to a decreasing and convex function of the geodesic distance are
attained at points of the dual regular $N$-gon. This partially solved the
problem about universal minima of $\omega^{\ast}_{N}$. The set of universal
minima of regular simplex $\omega_{d+1}^{\ast}$ was shown in [8] to be
$-\omega_{d+1}^{\ast}$.

During talk [6] given in January 2022 at ESI (available in the official
YouTube account of ESI) the author of this paper proved the following theorem:
given an $m$-stiff configuration $\omega_{N}$ (see Definition 4.1), its
universal minima are those and only those points on $S^{d}$ that form $m$
distinct dot products with points of $\omega_{N}$. We state this result in
Theorem 4.3111Theorem 4.3 was earlier announced in talk [5] in May 2021.. In
view of Remark 2.1 from the preliminaries, one can write

$\min\limits_{{\bf x}\in S^{d}}p_{f}({\bf
x},\omega_{N})=\sum\limits_{i=1}^{m}M_{i}f(2-2t_{i}),$ (8)

where $t_{1},\ldots,t_{m}$ are zeros of $P_{m}^{(d)}$,
$\\{\varphi_{1},\ldots,\varphi_{m}\\}$ is the correponding fundamental set of
polynomials, and $M_{i}=a_{0}(\varphi_{i})N$, $i=\overline{1,m}$.

During talk [6], we mentioned regular $2m$-gon on $S^{1}$, cube and cross-
polytope on $S^{d-1}$, and the $24$-cell on $S^{3}$ as examples of stiff
configurations. Since the duality of the cube and cross-polytope can be easily
seen as well as the duality of certain two isometric copies of the $24$-cell,
this theorem immediately implies the following. The set of universal minima of
a regular $2m$-gon on $S^{1}$ is the set of midpoints of the arcs joining its
neighboring points. The set of universal minima of a regular cross-polytope is
the dual cube and the one for a cube is the dual cross-polytope. The set of
universal minima of the $24$-cell is the dual $24$-cell. We state these
corollaries in Section 5.

The above mentioned theorem on stiff configurations is a straightforward
consequence of Lemma 3.5 in [4] (posted to ArXiv in March 2022). We state it
here as Lemma 6.3. Lemma 3.5 in [4] has three other similar immediate
consequences. All four follow if one just uses the well-known error formula
for Hermite interpolation (see, e.g., [20, Section 2.2] or [17, Theorem 4.5])
or its “slight” modifications to include interpolation at the endpoint(s)
(see, e.g., [15, Theorem 3.5.1]). One of those consequences222Mentioned in
abstract [7, p. 78] accepted on July 1, 2022. is that, for any strongly sharp
configuration $\omega_{N}$ (see Definition 4.4), the set of its universal
minima is the set of antipods of $\omega_{N}$. We state it in Theorem 4.5.
Therefore, the set of universal minima of a regular $(2m+1)$-gon is the set of
midpoints of the arcs joining neighboring vertices and the sets of universal
minima of the Schläffi configuration on $S^{5}$ and of the McLaughlin
configuration on $S^{21}$ are at the antipods of points of these
configurations (universal minima of regular simplex were mentioned earlier).

The remaining two consequences of [4, Lemma 3.5] are that universal maxima of
any strongly sharp or sharp antipodal configuration (see Definition 4.4) are
(only) at the points of the configuration itself, see [4]333The proof of this
was also given earlier in talk [6].. In particular, universal maxima of a
regular $N$-gon, regular simplex, regular cross-polytope, regular icosahedron,
and $E_{8}$ and Leech lattices are points of the configuration itself.

Dot products $t_{1},\ldots,t_{m}$ between distinct points in a strongly
$m$-sharp or $m$-sharp antipodal configuration are well-known (see Remark
2.2). The corresponding frequencies $M_{i}$ (independent of the point in
$\omega_{N}$) can be computed as explained in Section 2. So one can write

$\min\limits_{{\bf x}\in S^{d}}p_{f}({\bf
x},\omega_{N})=f(4)+\sum\limits_{i=1}^{m}M_{i}f(2+2t_{i}),$ (9)

for the minimum value for a strongly $m$-sharp configuration and

$\max\limits_{{\bf x}\in S^{d}}p_{f}({\bf
x},\omega_{N})=f(0)+\sum\limits_{i=1}^{m}M_{i}f(2-2t_{i}),$ (10)

for the maximum value for a strongly $m$-sharp or an $m$-sharp antipodal one,
where in the case of a sharp antipodal configuration, we have $t_{1}=-1$ and
$M_{1}=1$.

The work described above stimulated further research. The universal maxima of
the $600$-cell on $S^{3}$ (which is not sharp) were shown by Boyvalenkov,
Dragnev, Hardin, Saff, and Stoyanova [10] to be at points of the $600$-cell
itself (on ArXiv since July 15, 2022). A number of new stiff configurations
(which are also sharp) were found by Boyvalenkov, Dragnev, Hardin, Saff, and
Stoyanova [11] (on ArXiv since October 2022). The result of [6] implies that
their universal minima are at points of the corresponding dual configurations
(see Definition 4.1). Paper [11] further elaborates on each of these dual
configurations (as well as on antipods of strongly sharp configurations on
$S^{21}$ mentioned above).

For stiff or strongly sharp configurations considered in [11], values of
expressions in (8) or (9) are computed in [11, Table 2]. Results of [11,
Theorem 1.4] (see also [10]) provide lower and upper bounds for the potential
of any configuration $\omega_{N}$ on $S^{d}$ as soon as $\omega_{N}$ is at
least a $1$-design. When $\omega_{N}$ is stiff, strongly sharp, or sharp
antipodal, these bounds become sharp (one essentially gets equations (8)–(10)
mentioned above444In [11, Theorem 1.4] item (i) case $\epsilon=1$, dot
products $\alpha_{j}$ are zeros of
$(1+t)^{\epsilon}P_{k}^{(0,\epsilon)}(-t)$.). Proofs of Lemma 3.5 from [4] and
of Theorem 1.4 from [11] both use equivalent definition (6) of a spherical
design.

Some important regular configurations are not stiff or strongly sharp. The
author [3] (on ArXiv since October 9, 2022) found and characterized all
universal minima for the icosahedron (minima are only at vertices of the dual
dodecahedron), dodecahedron (dual icosahedron), and for the $E_{8}$ lattice on
$S^{7}$ ($2_{41}$ polytope). This was done by proving a general theorem for
$(2m-3)$-designs with vanishing $(2m-1)$-th and $2m$-th moments (the “skip one
add two” case). It was explicitly stated in talk [7]555The talk was given on
August 29, 2022. and its proof was earlier illustrated using the example of
icosahedron in talk [6]666Dodecahedron also fits the argument for icosahedron
from talk [6]..

One universal minimum and the universal minimum value of the potential of the
Leech lattice on $S^{23}$ was found by Boyvalenkov, Dragnev, Hardin, Saff, and
Stoyanova [11] (on ArXiv since October 31, 2022) who also proved a general
statement for the “skip one add two” case. Their theorem resorts to the
Gaussian-type quadrature for the corresponding polynomial space. It provides a
different (to a certain extent) proof for universal minima of icosahedron,
dodecahedron, and (without characterization of minima) $E_{8}$ lattice.

Critical points of the total potential of finite configurations of charges
were also analysed (see [2, 18] and references therein). This work is related
to the known Maxwell’s conjecture.

We remark that the problem about universal minima of the $600$-cell remains
open as well as the problem about universal maxima of stiff configurations
that are not sharp antipodal (e.g., cube).

In addition to Theorems 4.3 and 4.5, we find in this paper all universal
minima of six stiff configurations, for which universal minima have not been
characterized before, see Sections 8, 9, and 10.

Table 1 shows the current state of the art in finding universal extrema for
the classes of configurations of interest and for the $600$-cell. Table 2
summarizes results of our work in [3, 4, 6] and in this paper for particular
configurations. Last three lines of Table 3 contain the six stiff
configurations, for which we find here universal minima (the first three lines
follow immediately from Theorem 4.3).

Class of configurations | Universal maxima | Universal minima
---|---|---
Strongly sharp |  Solved [4] |  Theorem 4.5, announced [7], see also [11]
Sharp antipodal, non-stiff |  Solved [4] |  Solved [3, 11] for a subclass
Sharp antipodal and stiff |  Solved [4] |  Solved [6], see Theorem 4.3
Stiff, not sharp antipodal | Open |  Solved [6], see Theorem 4.3
$600$-cell |  Solved [10] | Open
Table 1: Current progress on the solution of the problem about universal
extrema of general classes of spherical configurations of interest and of the
$600$-cell.

When proving our main results, we use the Delsarte method (the Delsarte-Yudin
bound, linear programming method, or polynomial method), see the work by
Delsarte, Goethals, and Seidel [16]. This approach has been extensively used,
in particular, to find best-packing, kissing number, and minimal energy, as
well as to estimate the number of points in a spherical design (see, [21, 22,
23, 13, 24, 14, 9] and references therein).

##  4 General results

We first state the solution to Problem 1.1 for a class of spherical designs,
which we call here stiff configurations. Throughout the rest of the paper,
${\bf x}_{1},\ldots,{\bf x}_{N}$ will denote the points in a configuration
$\omega_{N}$ on $S^{d}$ whose total potential we are minimizing. For a given
$m\in\mathbb{N}$ and a given configuration $\omega_{N}\subset S^{d}$, denote
by $\mathcal{D}_{m}(\omega_{N})$ the set of all points ${\bf z}\in S^{d}$ for
which the set of dot products

$D({\bf z},\omega_{N}):=\\{{\bf z}\cdot{\bf x}_{i}:i=1,\ldots,N\\}$

has at most $m$ distinct elements. The set $D({\bf z},\omega_{N})$ is allowed
to contain $1$, which occurs when ${\bf z}\in\omega_{N}$. If
$\mathcal{D}_{m}(\omega_{N})\neq\emptyset$, then $\omega_{N}$ is contained in
a union of at most $m$ parallel hyperplanes.

######  Definition 4.1.

We call a point configuration $\omega_{N}\subset S^{d}$ $m$-stiff, $d,m\geq
1$, if $\omega_{N}$ is a spherical $(2m-1)$-design and the set
$\mathcal{D}_{m}(\omega_{N})$ is non-empty. The set
$\mathcal{D}_{m}(\omega_{N})$ of a given $m$-stiff configuration $\omega_{N}$
is called the dual configuration for $\omega_{N}$.

######  Remark 4.2.

If $\omega_{N}$ is an $m$-stiff configuration on $S^{d}$, $d,m\geq 1$, then
every point in $\mathcal{D}_{m}(\omega_{N})$ forms exactly $m$ distinct dot
products with points of $\omega_{N}$. If it did not, then $\omega_{N}$ would
be contained in $k<m$ parallel hyperplanes ${\bf x}\cdot{\bf v}=\alpha_{i}$,
$i=1,\ldots,k$, and the polynomial $u({\bf x}):=({\bf x}\cdot{\bf
v}-\alpha_{1})^{2}\cdots({\bf x}\cdot{\bf v}-\alpha_{k})^{2}$ of degree
$2k<2m-1$ would vanish on $\omega_{N}$ while having a positive integral over
$S^{d}$.

In view of this remark, the dual configuration for an $m$-stiff $\omega_{N}$
can be equivalently defined as the set of all points on $S^{d}$ that form
exactly $m$ distinct dot products with points of $\omega_{N}$.

There are many examples of stiff configurations other than the ones mentioned
when reviewing known results in Section 1. First, there are two interesting
examples on $S^{d-1}$ for $d\geq 3$ odd (both are $2$-stiff), two more on
$S^{5}$, and two more on $S^{6}$ (all are $3$-stiff). We describe these
configurations in Sections 8, 9, and 10. Furthermore, paper [11] analyses one
example on $S^{4}$ and a number of examples on higher-dimensional spheres.

We have the following result. It was prsented in talk [6]777Given in January
2022. with the proof.

######  Theorem 4.3.

Let $m\geq 1$, $d\geq 1$, $f:[0,4]\to(-\infty,\infty]$ be a function
continuous on $(0,4]$ with $f(0)=\lim\limits_{t\to 0^{+}}f(t)$, differentiable
on $(0,4)$ with a convex derivative $f^{(2m-2)}$ on $(0,4)$. If
$\omega_{N}=\\{{\bf x}_{1},\ldots,{\bf x}_{N}\\}$ is an $m$-stiff
configuration on the sphere $S^{d}$, then the potential

$p_{f}({\bf x},\omega_{N})=\sum\limits_{i=1}^{N}f(\left|{\bf x}-{\bf
x}_{i}\right|^{2}),\ \ \ {\bf x}\in S^{d},$

attains its absolute minimum over $S^{d}$ at every point of the set
$\mathcal{D}_{m}(\omega_{N})$.

If, in addition, $f^{(2m-2)}$ is strictly convex on $(0,4)$, then
$\mathcal{D}_{m}(\omega_{N})$ contains all points of absolute minimum of the
potential $p_{f}(\cdot,\omega_{N})$ on $S^{d}$.

Immediate consequences of Theorem 4.3 are stated in Section 5. No point from
an $m$-stiff configuration $\omega_{N}$ (or $-\omega_{N}$) can be in
$\mathcal{D}_{m}(\omega_{N})$, see Remark 2.1 or Lemma 6.2. Therefore, a point
from $\omega_{N}$ in Theorem 4.3 cannot be a minimizer of the potential of
$\omega_{N}$ when $f^{(2m-2)}$ is strictly convex.

The “slightly” less general case of Theorem 4.3 when $f$ has a positive
derivative $f^{(2m)}$ on $(0,4]$ is an important special case of our earlier
result [4, Lemma 3.5] restated here as Lemma 6.3. To see this, recall the
following well-known fact from a course in numerical analysis [20, Section
2.2] or [17, Theorem 4.5]. Let $h$ denote the Hermite interpolating polynomial
for $g(t)=f(2-2t)$ at the set of nodes $-1<t_{1}<\ldots<t_{m}<1$, which is
chosen to be the set of dot products that a point from
$\mathcal{D}_{m}(\omega_{N})$ forms with points from $\omega_{N}$. Then ${\rm
deg}\ \\!h\leq 2m-1$ and

$g(t)-h(t)=\frac{g^{(2m)}(c)}{(2m)!}(t-t_{1})^{2}\cdots(t-t_{m})^{2}\geq 0,\ \
\ t\in[-1,1),$ (11)

where $c=c(t)\in(-1,1)$. If $f(1)<\infty$, then (11) also holds for $t=1$. The
“full” case of Theorem 4.3 follows on applying Lemma 6.1 instead of (11), see
details in Section 6.

We consider next another class of regular configurations on the sphere, for
which universal minima can be found. If the set $\mathcal{D}_{m}(\omega_{N})$
contains a point from $-\omega_{N}$, then it is enough to require that
$\omega_{N}$ be a $(2m-2)$-design. These assumptions are satisfied by any
sharp configuration, which is an even order design. For the definition of a
sharp configuration, see [14].

######  Definition 4.4.

We call a point configuration $\omega_{N}\subset S^{d}$ $m$-sharp, $m\geq 1$,
if $\omega_{N}$ is a spherical $(2m-1)$-design and there are $m$ distinct
values of the dot product that can occur between any two distinct vectors from
$\omega_{N}$. We call an $m$-sharp configuration strongly $m$-sharp if it is a
spherical $2m$-design.

Examples of strongly sharp configurations are vertices of a regular
$(2m+1)$-gon inscribed in $S^{1}$, $m\in\mathbb{N}$, the set of vertices of a
regular $d$-simplex inscribed in $S^{d-1}$ ($m=1$), the Schläffi configuration
of $N=27$ points on $S^{5}$ ($m=2$), and the McLaughlin configuration of
$N=275$ points on $S^{21}$ ($m=2$). There are no strongly $m$-sharp
configurations on $S^{d}$ for $d\geq 2$ and $m\geq 3$. This follows from the
results of [16] and [1].

The classes of stiff and sharp configurations overlap. For example, sets of
vertices of a regular cross-polytope on $S^{d-1}$ and of a regular $2m$-gon
inscribed in $S^{1}$ are both stiff and sharp as well as the kissing
configuration $\overline{\omega}_{56}$ on $S^{6}$ and some examples mentioned
in [11, Table 2]. However, there are stiff configurations, which are not sharp
(e.g., the cube in $\mathbb{R}^{d}$ for $d\geq 3$ or the $24$-cell on
$S^{3}$). There are also sharp configurations, which are not stiff, e.g., the
regular $(2m+1)$-gon inscribed in $S^{1}$, the regular icosahedron on $S^{2}$,
and the minimal vectors of $E_{8}$ lattice on $S^{7}$. In addition, classes of
strongly sharp and stiff configurations are disjoint (see Proposition 7.3).

The following result888Mentioned in the abstract of [7, p. 78].  follows from
[4, Lemma 3.5] in the same way as Theorem 4.3.

######  Theorem 4.5.

Let $m\geq 1$, $d\geq 1$, $f:[0,4]\to(-\infty,\infty]$ be a function
continuous on $(0,4]$ with $f(0)=\lim\limits_{t\to 0^{+}}f(t)$, differentiable
on $(0,4)$ with a concave derivative $f^{(2m-1)}$ on $(0,4)$. If
$\omega_{N}=\\{{\bf x}_{1},\ldots,{\bf x}_{N}\\}\subset S^{d}$ is a strongly
$m$-sharp configuration, then the potential

$p_{f}({\bf x},\omega_{N})=\sum\limits_{i=1}^{N}f(\left|{\bf x}-{\bf
x}_{i}\right|^{2}),\ \ \ {\bf x}\in S^{d},$

attains its absolute minimum over $S^{d}$ at every point of $-\omega_{N}$.

If the concavity of $f^{(2m-1)}$ on $(0,4)$ is strict, then $-\omega_{N}$
contains all points of absolute minimum of the potential
$p_{f}(\cdot,\omega_{N})$ over $S^{d}$.

Name | $N$ | Sphere | Design | Sharpness | Stiffness | We
---|---|---|---|---|---|---
|  |  | strength |  |  | find
Regular $(2m+1)$-gon | $2m+1$ | $S^{1}$ | $2m$ |  strongly $m$-sharp | $-$ | both
Regular $2m$-gon | $2m$ | $S^{1}$ | $2m-1$ |  $m$-sharp |  $m$-stiff | both
Regular $d$-simplex | $d+1$ | $S^{d-1}$ | $2$ |  strongly $1$-sharp | $-$ | both
Regular cross-polytope ($d\geq 3$) | $2d$ | $S^{d-1}$ | $3$ |  $2$-sharp |  $2$-stiff | both
Cube ($d\geq 3$) | $2^{d}$ | $S^{d-1}$ | $3$ | $-$ |  $2$-stiff | min
Symmetrized $(2\ell+1)$-simplex | $4\ell+4$ | $S^{2\ell}$ | $3$ | $-$ |  $2$-stiff | min
Dual symm.$(2\ell\\!+\\!1)$-simpl., $\ell\\!\geq\\!2$ | $\binom{2\ell+2}{\ell+1}$ | $S^{2\ell}$ | $3$ | $-$ |  $2$-stiff | min
Icosahedron | $12$ | $S^{2}$ | $5$ |  $3$-sharp | $-$ | both
Dodecahedron | $20$ | $S^{2}$ | $5$ | $-$ | $-$ | min
$24$-cell | $24$ | $S^{3}$ | $5$ | $-$ |  $3$-stiff | min
Schläffi | $27$ | $S^{5}$ | $4$ |  strongly $2$-sharp | $-$ | both
Symmetrized Schläffi | $54$ | $S^{5}$ | $5$ | $-$ |  $3$-stiff | min
Roots of $E_{6}$ lattice | $72$ | $S^{5}$ | $5$ | $-$ |  $3$-stiff | min
Kissing | $56$ | $S^{6}$ | $5$ |  $3$-sharp |  $3$-stiff | both
$E_{7}$ lattice | $126$ | $S^{6}$ | $5$ | $-$ |  $3$-stiff | min
$E_{8}$ lattice | $240$ | $S^{7}$ | $7$ |  $4$-sharp | $-$ | both
McLaughlin | $275$ | $S^{21}$ | $4$ |  strongly $2$-sharp | $-$ | both
Leech lattice | $196560$ | $S^{23}$ | $11$ |  $6$-sharp | $-$ | max

Table 2: Summary of our results in [3, 4, 6] and in this paper giving all
universal extrema of certain configurations (see the last column). Entry with
a dash indicates that the property does not hold. We remark that all universal
maxima of the $600$-cell were found in [10] and one universal minimum was
found in [11] for the kissing $56$-point configuration on $S^{6}$ and for
Leech lattice on $S^{23}$. Paper [11] studies universal minima for a number of
stiff configurations (which are also sharp) not listed in this table.

.

Sphere | Pairs of mutually dual stiff configurations
---|---
$S^{1}$ |  regular $2m$-gon |  regular $2m$-gon rotated by $\frac{\pi}{2m}$ rad
$S^{d-1}$, $d\geq 3$ |  regular cross-polytope $\omega^{\ast}_{2d}$ |  cube $U_{d}$
$S^{3}$ |  $24$-cell $\overline{\omega}_{24}$ |  $24$-cell $\widetilde{\omega}_{24}$
$S^{d-1}$, $d\geq 3$ odd |  symmetrized regular $d$-simplex $\overline{\omega}_{2d+2}$ |  $\overline{\omega}_{\kappa(d)}$ defined by (17)
$S^{5}$ |  symmetrized Schläffi configuration $\overline{\omega}_{54}$ |  root system of $E_{6}$-lattice $\overline{\omega}_{72}$
$S^{6}$ |  kissing configuration $\overline{\omega}_{56}$ |  minimal vectors of $E_{7}$-lattice $\overline{\omega}_{126}$
Table 3: Pairs of mutually dual stiff configurations. Universal minima for the
first three lines are immediate consequences of Theorem 4.3 proved in talk
[6]. Universal minima of configurations in the last three lines are obtained
in Sections 8, 9, and 10.

Theorem 4.5 when $g(t)=f(2-2t)$ has a positive derivative $g^{(2m+1)}$ on
$[-1,1)$ can be derived from [4, Lemma 3.5] by recalling the following fact
from a numerical analysis course [15, Theorem 3.5.1]. If $h$ is the polynomial
of degree at most $2m$ interpolating $g$ to the second order at the nodes
$-1<t_{1}<\ldots<t_{m}<1$ and to the first order at $t_{0}=-1$, then

$g(t)-h(t)=\frac{g^{(2m+1)}(c)}{(2m+1)!}(t-t_{1})^{2}\cdots(t-t_{m})^{2}(t+1)\geq
0,\ \ \ t\in[-1,1),$ (12)

where $c=c(t)\in(-1,1)$. Numbers $t_{1},\ldots,t_{m}$ are chosen to be
opposite to dot products between distinct points in $\omega_{N}$. If
$f(1)<\infty$, then (12) also holds for $t=1$. The “full” case of Theorem 4.5
follows on applying Lemma 6.1 instead of (12), see details in Section 6.

Note that in the proofs of Theorems 4.3 and 4.5, we do not need to show the
positive definiteness of the interpolating polynomial when utilizing the
linear programming approach of [16, 14].

Other main results of this paper are Theorems 8.5, 9.3, and 10.3.

##  5 Immediate consequences of Theorems 4.3 and 4.5

This section discusses universal minima given by Theorem 4.3 for regular
$2m$-gon, regular cross-polytope, cube, and $24$-cell (finding the dual for
each configuration is elementary) and universal minima given by Theorem 4.5
for a regular $(2m+1)$-gon, regular simplex, and Schläffi and McLaughlin
configurations. For power potentials (3), the first five lines of Table 4 were
proved in [28, 29, 25, 26]. For regular simplex, universal minima were found
in [8] (here we provide an alternative proof).

The following auxiliary statements give dual configurations. Let
$\widetilde{\omega}_{N}$ denote the set of midpoints of arcs on $S^{1}$ that
join the neighboring points in the regular $N$-gon $\omega^{\ast}_{N}$
inscribed in $S^{1}$.

######  Lemma 5.1.

For $N=2m$, $m\geq 1$, we have
$\mathcal{D}_{m}(\omega_{N}^{\ast})=\widetilde{\omega}_{N}$.

###### Proof.

Every point from $\widetilde{\omega}_{N}$, $N=2m$, forms $m$ distinct dot
products with points of $\omega_{N}^{\ast}$, while every point from
$\omega_{N}^{\ast}$ itself forms $m+1$ dot products and every other point on
$S^{1}$ forms $2m$ dot products with points of $\omega_{N}^{\ast}$. ∎

######  Lemma 5.2.

For every $d\geq 2$, we have $\mathcal{D}_{2}(\omega_{2d}^{\ast})=U_{d}$.

###### Proof.

The inclusion $U_{d}\subset\mathcal{D}_{2}(\omega_{2d}^{\ast})$ is immediate.
Let $x_{i}$ be a non-zero coordinate of ${\bf
x}:=(x_{1},\ldots,x_{d})\in\mathcal{D}_{2}(\omega_{2d}^{\ast})$. Then ${\bf
x}\cdot(\pm{\bf e}_{i})=\pm x_{i}$. Every other coordinate $x_{j}$ must have
the same absolute value as $x_{i}$ (otherwise $\pm x_{j}$ would give extra dot
product(s)). Then $\left|x_{i}\right|=\frac{1}{\sqrt{d}}$, $i=\overline{1,d}$;
that is, ${\bf x}\in U_{d}$. ∎

######  Lemma 5.3.

For every $d\geq 2$, we have $\mathcal{D}_{2}(U_{d})=\omega_{2d}^{\ast}$.

###### Proof.

The inclusion $\omega_{2d}^{\ast}\subset\mathcal{D}_{2}(U_{d})$ is immediate.
If ${\bf x}=(x_{1},\ldots,x_{d})\in\mathcal{D}_{2}(U_{d})$ had at least two
non-zero coordinates, say $x_{1}$ and $x_{2}$, then dot product ${\bf
x}\cdot\left(\pm\frac{1}{\sqrt{d}},\pm\frac{1}{\sqrt{d}},\frac{1}{\sqrt{d}},\ldots,\frac{1}{\sqrt{d}}\right)=\frac{\pm
x_{1}\pm
x_{2}}{\sqrt{d}}+\frac{x_{3}}{\sqrt{d}}+\ldots+\frac{x_{d}}{\sqrt{d}}$ would
have at least three distinct values. Therefore, only one coordinate of ${\bf
x}$ is non-zero; i.e., ${\bf x}\in\omega_{2d}^{\ast}$. ∎

The $24$-cell $\overline{\omega}_{24}\subset S^{3}$ is the set of $N=24$
points with two coordinates equal $0$ and each of the remaining two
coordinates being $1/\sqrt{2}$ or $-1/\sqrt{2}$. The $24$-cell is known to be
a $5$-design. At the same time, every vector $\pm{\bf e}_{i}$,
$i=\overline{1,4}$, where $\\{{\bf e}_{1},{\bf e}_{2},{\bf e}_{3},{\bf
e}_{4}\\}$ is the standard basis in $\mathbb{R}^{4}$, forms only $3$ distinct
dot products with vectors of the $24$-cell, which are $\pm 1/\sqrt{2}$ and
$0$. Furthermore, each of the $16$ vectors of the form
$\left(\pm\frac{1}{2},\pm\frac{1}{2},\pm\frac{1}{2},\pm\frac{1}{2}\right)$
forms these three values of the dot product with vectors from
$\overline{\omega}_{24}$ as well. Therefore, $\overline{\omega}_{24}$ is
$3$-stiff. Vectors $\pm{\bf e}_{i}$, $i=\overline{1,4},$ and
$\left(\pm\frac{1}{2},\pm\frac{1}{2},\pm\frac{1}{2},\pm\frac{1}{2}\right)$
form another $24$-cell denoted by $\widetilde{\omega}_{24}$.

Configuration $\omega_{N}$ | Sphere | Assumption | Absolute minima
---|---|---|---
Regular $2m$-gon | $S^{1}$ |  $f^{(2m-2)}$ is convex |  regular $2m$-gon rotated by $\frac{\pi}{2m}$ rad
Regular $(2m+1)$-gon | $S^{1}$ |  $f^{(2m-1)}$ is concave |  antipod of regular $(2m+1)$-gon
Regular $d$-simplex $\omega_{d+1}^{\ast}$ | $S^{d-1}$ |  $f^{\prime}$ is concave |  regular $d$-simplex $-\omega^{\ast}_{d+1}$
Cross-polytope $\omega^{\ast}_{2d}$, $d\geq 3$ | $S^{d-1}$ |  $f^{\prime\prime}$ is convex |  cube $U_{d}$
Cube $U_{d}$, $d\geq 3$ | $S^{d-1}$ |  $f^{\prime\prime}$ is convex |  cross-polytope $\omega^{\ast}_{2d}$
$24$-cell $\overline{\omega}_{24}$ | $S^{3}$ |  $f^{(4)}$ is convex |  $24$-cell $\widetilde{\omega}_{24}$
Schläffi $\overline{\omega}_{27}$ | $S^{5}$ |  $f^{\prime\prime\prime}$ is concave | $-\overline{\omega}_{27}$
McLaughlin $\overline{\omega}_{275}$ | $S^{21}$ |  $f^{\prime\prime\prime}$ is concave | $-\overline{\omega}_{275}$
Table 4: Immediate consequences of Theorems 4.3 and 4.5. We assume that
$f:[0,4]\to(-\infty,\infty]$ is continuous on $(0,4]$ with
$f(0)=\lim\limits_{t\to 0^{+}}f(t)$ and differentiable on $(0,4)$ with an
additional assumption on $f$ specified on $(0,4)$ in the third column. Then
the potential $p_{f}(\cdot,\omega_{N})$ attains its absolute minimum over the
specified sphere at points of the set in the fourth column. If the assumption
in the third column holds strictly, then no other absolute minima exist.

######  Lemma 5.4.

$\mathcal{D}_{3}(\overline{\omega}_{24})=\widetilde{\omega}_{24}$.

###### Proof.

The inclusion
$\widetilde{\omega}_{24}\subset\mathcal{D}_{3}(\overline{\omega}_{24})$ is
immediate. Let ${\bf
x}=(x_{1},x_{2},x_{3},x_{4})\in\mathcal{D}_{3}(\overline{\omega}_{24})$. There
are at most three distinct numbers among all dot products $\frac{\pm x_{i}\pm
x_{j}}{\sqrt{2}}$, $1\leq i\neq j\leq 4$. Then all non-zero coordinates of
${\bf x}$ must have the same absolute value (denoted by $a$). If ${\bf x}$ had
two or three non-zero coordinates, it would form dot products
$0,\pm\frac{a}{\sqrt{2}},\pm\sqrt{2}a$ with vectors from
$\overline{\omega}_{24}$. Therefore, ${\bf x}$ has one non-zero coordinate
($a=1$) or all its coordinates equal $\pm a$ ($a=1/2$); i.e., ${\bf
x}\in\widetilde{\omega}_{24}$. ∎

Immediate consequences of Theorems 4.3 and 4.5 are listed in Table 4.

##  6 Proofs of Theorems 4.3 and 4.5

We first establish the following interpolation lemma for the potential
function. It is a “slight” modification of [4, Lemma 3.1]. The case $\nu=0$
corresponds to stiff configurations and Theorem 4.3. The case $\nu=1$
corresponds to strongly sharp configurations and Theorem 4.5.

######  Lemma 6.1.

Suppose $\nu=0$ or $1$. Let $n\geq 1+\nu$, $L:=2n-2-\nu$, and $-1\leq
t_{1}<\ldots<t_{n}<1$ be arbitrary nodes such that $t_{1}>-1$ if $\nu=0$ and
$t_{1}=-1$ if $\nu=1$. Suppose also that $g:[-1,1]\to(-\infty,\infty]$ is a
function continuous on $[-1,1)$ with $g(1)=\lim\limits_{t\to 1^{-}}g(t)$ and
differentiable on $(-1,1)$ with the derivative $g^{(L)}$ being convex on
$(-1,1)$. Let $q\in\mathbb{P}_{L+1}$ be the polynomial such that

$q(t_{i})=g(t_{i}),\ \ i=1,\ldots,n,\ \ \text{and}\ \
q^{\prime}(t_{i})=g^{\prime}(t_{i}),\ \ i=1+\nu,\ldots,n.$ (13)

Then

$q(t)\leq g(t),\ \ t\in[-1,1].$ (14)

If, in addition, the derivative $g^{(L)}$ is strictly convex on $(-1,1)$, then
the inequality in (14) is strict for
$t\in[-1,1]\setminus\\{t_{1},\ldots,t_{n}\\}$.

###### Proof.

The assertion of the lemma follows immediately when $L=0$; i.e., when $n=1$
and $\nu=0$. Therefore, we assume that $L\geq 1$ (then $n\geq 2$). Inequality
(14) holds trivially for $t=t_{1},\ldots,t_{n}$ (as equality) and, if
$g(1)=\infty$, at $t=1$ (as a strict inequality). Therefore, we choose
arbitrary $x\in[-1,1]\setminus\\{t_{1},\ldots,t_{n}\\}$ assuming additionally
that $x<1$ if $g(1)=\infty$. Denote

$h(t):=q(t)+b(t-t_{1})^{2-\nu}\prod_{i=2}^{n}(t-t_{i})^{2},$

where the constant $b$ is chosen so that $h(x)=g(x)$. Observe that the
polynomial $h$ also satisfies interpolation conditions (13). The function
$\tau(t):=g(t)-h(t)$ has at least $n+1$ distinct zeros in $[-1,1]$ at least
$n-\nu$ of which have multiplicity two and are located in $(-1,1)$. By the
Rolle’s theorem, the derivative $\tau^{\prime}$ has at least $2n-\nu=L+2$
distinct zeros in $(-1,1)$, which implies that the derivative $\tau^{(L)}$ has
at least three distinct zeros in $(-1,1)$. The polynomial $h^{(L)}$ has degree
at most two with the coefficient of $t^{2}$ being $\frac{(L+2)!}{2}b$. It
interpolates the convex function $g^{(L)}$ at at least three distinct points
in $(-1,1)$. Then $b\geq 0$. Since $(x-t_{1})^{2-\nu}>0$ both for $\nu=0$ and
$\nu=1$, we have

$g(x)=h(x)=q(x)+b(x-t_{1})^{2-\nu}\prod_{i=2}^{n}(x-t_{i})^{2}\geq q(x),$ (15)

which proves (14). If $g^{(L)}$ is strictly convex on $(-1,1)$, then $b\neq 0$
(otherwise $g^{(L)}$ would be interpolated by the polynomial $h^{(L)}$ of
degree at most one at three distinct points). Therefore, $b>0$ and the
inequality in (14) is strict on $[-1,1]\setminus\\{t_{1},\ldots,t_{n}\\}$. ∎

######  Lemma 6.2.

Let $\omega_{N}$ be an $m$-stiff configuration on $S^{d}$, $d,m\geq 1$. Then
there is no point in $\mathcal{D}_{m}(\omega_{N})$ that forms dot products $1$
or $-1$ with points from $\omega_{N}$. Let $\omega_{N}$ be strongly $m$-sharp.
Then $\omega_{N}$ contains no antipodal pair and
$\mathcal{D}_{m}(\omega_{N})=\emptyset$.

###### Proof.

Assume to the contrary that there is a point ${\bf
y}\in\mathcal{D}_{m}(\omega_{N})$ such that ${\bf y}\cdot{\bf x}_{\ell}=1$ or
$-1$ for some ${\bf x}_{\ell}\in\omega_{N}$. Let $-1\leq
t_{1}<\ldots<t_{k}\leq 1$, $k\leq m$, be the distinct dot products ${\bf y}$
forms with vectors from $\omega_{N}$. If ${\bf y}\cdot{\bf x}_{\ell}=1$, then
$t_{k}=1$ and we let $p(t):=(t-t_{1})^{2}\cdots(t-t_{k-1})^{2}(1-t)$. If ${\bf
y}\cdot{\bf x}_{\ell}=-1$, then $t_{1}=-1$ and we let
$p(t):=(t+1)(t-t_{2})^{2}\cdots(t-t_{k})^{2}$. Define $u({\bf x}):=p({\bf
x}\cdot{\bf y})$, ${\bf x}\in\mathbb{R}^{d+1}$. Then $u$ has degree at most
$2m-1$, $u({\bf x}_{i})=0$, $i=1,\ldots,N$, and $u({\bf x})>0$ for
$\sigma_{d}$-almost all ${\bf x}\in S^{d}$. Then, since $\omega_{N}$ is a
$(2m-1)$-design,

$0=\frac{1}{N}\sum\limits_{i=1}^{N}u({\bf x}_{i})=\int_{S^{d}}u({\bf x})\
\\!d\sigma_{d}({\bf x})>0.$

This contradiction proves the first part of the lemma.

Let now $\omega_{N}$ be strongly $m$-sharp. Assume to the contrary that
$\omega_{N}$ contains an antipodal pair ${\bf x}_{\ell}=-{\bf x}_{i}$. Let
$-1=t_{1}<t_{2}<\ldots<t_{m}<1$ be dot products between distinct points in
$\omega_{N}$. Letting $p(t):=(t+1)(t-t_{2})^{2}\cdots(t-t_{m})^{2}(1-t)$ and
$u({\bf x}):=p({\bf x}\cdot{\bf x}_{\ell})$ and using a similar argument, we
arrive at a contradiction proving that $\omega_{N}$ has no antipodal pair.

Finally, assume to the contrary that there is a point ${\bf
z}\in\mathcal{D}_{m}(\omega_{N})$. Let $-1\leq t_{1}<\ldots<t_{k}\leq 1$,
$k\leq m$, be distinct elements in $D({\bf z},\omega_{N})$. Letting
$p(t):=(t-t_{1})^{2}\cdots(t-t_{k})^{2}$ and $u({\bf x}):=p({\bf x}\cdot{\bf
z})$ and using a similar argument again, we arrive at a contradiction proving
that $\mathcal{D}_{m}(\omega_{N})$ is empty. ∎

Replacing $g$ and $q$ with $-g$ and $-q$ in [4, Lemma 3.5] we obtain the
following.

######  Lemma 6.3.

Suppose that $d,n\geq 1$ and that a function $g:[-1,1]\to(-\infty,\infty]$ and
a set $A$ consisting of numbers $-1\leq t_{1}<\ldots<t_{n}\leq 1$ are
arbitrary. Suppose also that $q$ is a polynomial such that
$q(t_{i})=g(t_{i})$, $i=1,\ldots,n$, and

$q(t)\leq g(t),\ \ t\in[-1,1].$ (16)

If $\omega_{N}=\\{{\bf x}_{1},\ldots,{\bf x}_{N}\\}\subset S^{d}$ is a
spherical $M$-design, where $M\geq{\rm deg}\ \\!q$, and ${\bf y}\in S^{d}$ is
any point such that $D({\bf y},\omega_{N})\subset A$, then the potential

$p^{g}({\bf x},\omega_{N}):=\sum\limits_{i=1}^{N}g({\bf x}\cdot{\bf x}_{i}),\
\ \ {\bf x}\in S^{d},$

attains its absolute minimum over $S^{d}$ at point ${\bf y}$.

If, in addition, the inequality in (16) is strict for every
$t\in[-1,1]\setminus A$, then any point ${\bf z}\in S^{d}$ such that $D({\bf
z},\omega_{N})\not\subset A$ is not a point of absolute minimum of $p^{g}(\
\\!\cdot\ \\!,\omega_{N})$ on $S^{d}$.

We are now ready to prove Theorems 4.3 and 4.5.

###### Proof of Theorem 4.3.

Let ${\bf y}$ be any point from $\mathcal{D}_{m}(\omega_{N})$. Let
$-1<\tau_{1}<\ldots<\tau_{m}<1$ be distinct elements of the set $D({\bf
y},\omega_{N})=\\{{\bf y}\cdot{\bf x}_{i}:1\leq i\leq N\\}$. There are exactly
$m$ dot products there in view of Remark 4.2 with $\tau_{1}>-1$ and
$\tau_{m}<1$ in view of Lemma 6.2. We apply Lemma 6.1 with $\nu=0$, $n=m$, and
$g(t)=f(2-2t)$ so that $g^{(L)}$ is convex on $(-1,1)$, where $L=2m-2$. Then
the Hermite interpolating polynomial $q\in\mathbb{P}_{2m-1}$ for $g$ at the
nodes $\tau_{i}$ satisfies $q(t)\leq g(t)$, $t\in[-1,1]$. Since $\omega_{N}$
is a spherical ($2m-1$)-design and $D({\bf
y},\omega_{N})=\\{\tau_{1},\ldots,\tau_{m}\\}$, by Lemma 6.3 (with $n=m$ and
$M=2m-1$), the potential $p_{f}(\cdot,\omega_{N})=p^{g}(\cdot,\omega_{N})$
attains its absolute minimum over $S^{d}$ at ${\bf y}$.

If $f^{(2m-2)}$ is strictly convex on $(0,4)$, then so is $g^{(2m-2)}$ on
$(-1,1)$ and, by Lemma 6.1, $q(t)<g(t)$,
$t\in[-1,1]\setminus\\{\tau_{1},\ldots,\tau_{m}\\}$. If ${\bf z}\in S^{d}$ is
any point not in $\mathcal{D}_{m}(\omega_{N})$, then the set $D({\bf
z},\omega_{N})$ contains at least $m+1$ distinct dot products. Hence, $D({\bf
z},\omega_{N})\not\subset\\{\tau_{1},\ldots,\tau_{m}\\}$. In view of Lemma
6.3, the point ${\bf z}$ is not a point of absolute minimum of
$p^{g}(\cdot,\omega_{N})$ on $S^{d}$, and, hence, of
$p_{f}(\cdot,\omega_{N})$. Thus, $\mathcal{D}_{m}(\omega_{N})$ is the set of
all points of absolute minimum of $p_{f}(\cdot,\omega_{N})$ on $S^{d}$. ∎

###### Proof of Theorem 4.5.

Let $-1<\tau_{1}<\ldots<\tau_{m}<\tau_{m+1}=1$ be the distinct values of the
dot product between points in $\omega_{N}$. Here, we have $\tau_{1}>-1$ in
view of Lemma 6.2. We apply Lemma 6.1 with $\nu=1$, $n=m+1$,
$t_{1}=-\tau_{m+1}=-1$, $t_{2}=-\tau_{m}$, …, $t_{m+1}=-\tau_{1}$, and
$g(t)=f(2-2t)$ so that $g^{(L)}$ is convex on $(-1,1)$, where $L=2m-1$. Then
the polynomial $q\in\mathbb{P}_{2m}$, which interpolates both $g$ and
$g^{\prime}$ at $t_{2},\ldots,t_{m+1}$ and only $g$ at $t_{1}$ satisfies
$q(t)\leq g(t)$, $t\in[-1,1]$. Choose any point ${\bf y}$ in $-\omega_{N}$.
Then $D({\bf y},\omega_{N})\subset\\{t_{1},\ldots,t_{m+1}\\}$. Since
$\omega_{N}$ is a $2m$-design, by Lemma 6.3 (with $n=m+1$ and $M=2m$) the
potential $p_{f}(\cdot,\omega_{N})=p^{g}(\cdot,\omega_{N})$ attains its
absolute minimum over $S^{d}$ at ${\bf y}$.

Let now $f^{(2m-1)}$ be strictly concave on $(0,4)$. Then $g^{(2m-1)}$ is
strictly convex on $(-1,1)$. Let ${\bf z}$ be an arbitrary point in
$S^{d}\setminus(-\omega_{N})$. Assume to the contrary that $D({\bf
z},\omega_{N})\subset\\{t_{1},\ldots,t_{m+1}\\}$. If $D({\bf z},\omega_{N})$
did not contain $t_{1}=-1$, then it would contain at most $m$ distinct
elements making the set $\mathcal{D}_{m}(\omega_{N})$ non-empty. By Lemma 6.2,
this is not possible. Consequently, $D({\bf z},\omega_{N})$ contains $-1$.
Then ${\bf z}\in-\omega_{N}$ contradicting its choice. Thus, $D({\bf
z},\omega_{N})\not\subset\\{t_{1},\ldots,t_{m+1}\\}$. By Lemma 6.1, we have
$q(t)<g(t)$, $t\in[-1,1]\setminus\\{t_{1},\ldots,t_{m+1}\\}$. Then by Lemma
6.3, the point ${\bf z}$ is not a point of absolute minimum of the potential
$p_{f}(\cdot,\omega_{N})=p^{g}(\cdot,\omega_{N})$ on $S^{d}$. ∎

##  7 Dot product distribution related to stiff configurations

In this section, we study the distribution of dot products between vectors
from $\mathcal{D}_{m}(\omega_{N})$ and $\omega_{N}$. Spherical designs can be
characterized using Gegenbauer polynomials as follows.

######  Theorem 7.1.

(see [16] or [9, Theorems 5.2.2 and 5.4.2]) Let $d,n\geq 1$ and
$\omega_{N}=\\{{\bf x}_{1},\ldots,{\bf x}_{N}\\}$ be a point configuration on
$S^{d}$. The following are equivalent:

  * (i)

$\omega_{N}$ is a spherical $n$-design;

  * (ii)

$\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{N}P_{k}^{(d)}({\bf x}_{i}\cdot{\bf
x}_{j})=0,\ \ k=1,\ldots,n;$

  * (iii)

For every polynomial $q\in\mathbb{P}_{n}$, we have $p^{q}({\bf
y},\omega_{N}):=\sum_{i=1}^{N}q({\bf y}\cdot{\bf x}_{i})=a_{0}(q)N$, ${\bf
y}\in S^{d}$, where

$a_{0}(q):=\int_{-1}^{1}q(t)w_{d}(t)\ \\!dt$

is the $0$-th Gegenbauer coefficient of polynomial $q$.

One can use Theorem 7.1 part (ii) to find the strength of every spherical
design mentioned in this paper. We next restate Remark 2.1 formally and, for
completeness, provide its proof. Here, $\varphi_{1},\ldots,\varphi_{m}$ are
the fundamental polynomials for set of nodes (4); that is,
$\varphi_{i}\in\mathbb{P}_{m-1}$, $\varphi_{i}(\kappa_{i}^{m})=1$, and
$\varphi_{i}(\kappa_{j}^{m})=0$, $j\neq i$, $i=1,\ldots,m$.

######  Proposition 7.2.

If $\omega_{N}$ is an $m$-stiff configuration on $S^{d}$, then for every ${\bf
z}\in\mathcal{D}_{m}(\omega_{N})$, the set $D({\bf z},\omega_{N})$ contains
exactly $m$ distinct elements located in $(-1,1)$, which are the zeros
$\kappa_{1}^{m},\ldots,\kappa_{m}^{m}$ of Gegenbauer polynomial $P_{m}^{(d)}$.
Furthermore, the number of indices $i$ such that ${\bf z}\cdot{\bf
x}_{i}=\kappa_{j}^{m}$ does not depend on ${\bf z}$ and equals
$a_{0}(\varphi_{j})N$, $j=1,\ldots,m$.

In particular, if $m=2$, then for every ${\bf
z}\in\mathcal{D}_{2}(\omega_{N})$, we have $D({\bf
z},\omega_{N})=\left\\{-\frac{1}{\sqrt{d+1}},\frac{1}{\sqrt{d+1}}\right\\}$,
and, if $m=3$, then for every ${\bf z}\in\mathcal{D}_{3}(\omega_{N})$, we have
$D({\bf z},\omega_{N})=\left\\{\pm\sqrt{\frac{3}{d+3}},0\right\\}$.

###### Proof.

Denote by $-1\leq t_{1}<\ldots<t_{m}\leq 1$ the distinct elements in $D({\bf
z},\omega_{N})$ (in view of Remark 4.2, there are exactly $m$ of them). Let
$M_{j}:=\\#\\{i:{\bf z}\cdot{\bf x}_{i}=t_{j}\\}$, $j=1,\ldots,m$. Denote
$P(t):=(t-t_{1})\cdots(t-t_{m})$. Since $\omega_{N}$ is a $(2m-1)$-design,
using item (iii) of Theorem 7.1, for every polynomial $p\in\mathbb{P}_{m-1}$,
we have

$a_{0}(pP)=\frac{1}{N}\sum\limits_{i=1}^{N}p({\bf z}\cdot{\bf x}_{i})P({\bf
z}\cdot{\bf x}_{i})=\frac{1}{N}\sum\limits_{j=1}^{m}M_{j}p(t_{j})P(t_{j})=0.$

Then $P\bot\ \\!\mathbb{P}_{m-1}$ on $[-1,1]$ with weight $w_{d}$; that is,
$P$ is a constant multiple of the Gegenbauer polynomial $P_{m}^{(d)}$. Then
$t_{j}=\kappa_{j}^{m}$, $j=1,\ldots,m$, and we have

$a_{0}(\varphi_{k})=\frac{1}{N}\sum\limits_{i=1}^{N}\varphi_{k}({\bf
z}\cdot{\bf
x}_{i})=\frac{1}{N}\sum\limits_{j=1}^{m}M_{j}\varphi_{k}(\kappa_{j}^{m})=\frac{M_{k}}{N};$

that is, $M_{k}=a_{0}(\varphi_{k})N$, $k=1,\ldots,m$.

Since $P_{2}^{(d)}(t)=\frac{d+1}{d}t^{2}-\frac{1}{d}$ and
$P_{3}^{(d)}(t)=\frac{d+3}{d}t^{3}-\frac{3}{d}t$, the sets of zeros of these
polynomials give the set $D({\bf z},\omega_{N})$ for $m=2$ and $3$. ∎

We conclude this section with the following consequence of Lemma 6.2.

######  Proposition 7.3.

The classes of stiff and strongly sharp configurations on $S^{d}$, $d\geq 1$,
are disjoint.

###### Proof of Proposition 7.3.

Assume to the contrary that there exists a configuration $\omega_{N}\subset
S^{d}$ which is $k$-stiff and strongly $m$-sharp for some $k,m\geq 1$. By
Lemma 6.2, the set $\mathcal{D}_{m}(\omega_{N})$ is empty. Then $k\geq m+1$.
Since $\omega_{N}$ is $k$-stiff, it must be at least a $(2m+1)$-design. If
$\omega_{N}$ is not $(m+1)$-stiff, then, by definition,
$\mathcal{D}_{m+1}(\omega_{N})=\emptyset$. If $\omega_{N}$ is $(m+1)$-stiff,
then by Lemma 6.2, the set $\mathcal{D}_{m+1}(\omega_{N})$ does not contain
any point from $\omega_{N}$. However, since $\omega_{N}$ is strongly
$m$-sharp, every point from $\omega_{N}$ is in
$\mathcal{D}_{m+1}(\omega_{N})$. This contradiction proves the proposition. ∎

##  8 Symmetrized regular simplex and its dual

In this section, we describe universal minima for the first pair of stiff
configurations. Let $d\geq 3$ be odd. Denote
$\overline{\omega}_{2d+2}=\omega^{\ast}_{d+1}\cup(-\omega^{\ast}_{d+1})$,
where $\omega^{\ast}_{d+1}=\\{{\bf x}_{0},{\bf x}_{1}\ldots,{\bf x}_{d}\\}$ is
the set of vertices of a regular $d$-simplex inscribed in $S^{d-1}$. Since
$\omega_{d+1}^{\ast}$ is a spherical $2$-design, the symmetrized regular
simplex $\overline{\omega}_{2d+2}$ is a spherical $3$-design. Denote
$\kappa(d):=\binom{d+1}{\frac{d+1}{2}}$ and let

$\overline{\omega}_{\kappa(d)}:=\left\\{\frac{2\sqrt{d}}{d+1}\sum\limits_{i\in
I}{\bf x}_{i}:I\subset\\{0,1,\ldots,d\\},\ \\#I=\frac{d+1}{2}\right\\}.$ (17)

It is not difficult to see that $\overline{\omega}_{\kappa(d)}\subset
S^{d-1}$. By Lemma 8.2 below,
$\mathcal{D}_{2}(\overline{\omega}_{2d+2})=\overline{\omega}_{\kappa(d)}$.
Thus, $\overline{\omega}_{2d+2}$ is $2$-stiff. The configuration
$\overline{\omega}_{\kappa(d)}$, in turn, is a spherical $3$-design (see Lemma
8.3 below) with
$\mathcal{D}_{2}(\overline{\omega}_{\kappa(d)})=\overline{\omega}_{2d+2}$ in
view of Lemma 8.4. Then $\overline{\omega}_{\kappa(d)}$ is also $2$-stiff. We
remark that for $d=2$, the configuration $\overline{\omega}_{2d+2}$ is a
regular hexagon on $S^{1}$, which is $3$-stiff, and its dual is also a regular
hexagon.

######  Proposition 8.1.

For $d\geq 4$ even, $\mathcal{D}_{2}(\overline{\omega}_{2d+2})=\emptyset$ and
$\overline{\omega}_{2d+2}$ is not stiff.

###### Proof.

The configuration $\overline{\omega}_{2d+2}$ is a $3$-design. Assume to the
contrary that $\mathcal{D}_{2}(\overline{\omega}_{2d+2})\neq\emptyset$. Then
$\overline{\omega}_{2d+2}$ is $2$-stiff. By Proposition 7.2, there is a point
${\bf x}\in S^{d-1}$ such that ${\bf x}\cdot{\bf
x}_{i}=\pm\frac{1}{\sqrt{d}}$, $i=0,1,\ldots,d$. Let $k$ be the number of
indices $i$ such that ${\bf x}\cdot{\bf x}_{i}=\frac{1}{\sqrt{d}}$. Then,
since $d+1$ is odd,

${\bf x}\cdot\sum_{i=0}^{d}{\bf x}_{i}=\sum\limits_{i=0}^{d}{\bf x}\cdot{\bf
x}_{i}=\frac{k}{\sqrt{d}}-\frac{d+1-k}{\sqrt{d}}=\frac{2k-(d+1)}{\sqrt{d}}\neq
0$

contradicting the fact that the center of mass of $\omega_{d+1}^{\ast}$ is at
the origin. Therefore, $\mathcal{D}_{2}(\overline{\omega}_{2d+2})=\emptyset$,
and $\overline{\omega}_{2d+2}$ cannot be $1$\- or $2$-stiff. Since $d\geq 4$,
$\overline{\omega}_{2d+2}$ is not a $4$-design. Then
$\overline{\omega}_{2d+2}$ cannot be $m$-stiff for any $m\geq 3$. ∎

######  Lemma 8.2.

For every $d\geq 3$ odd,
$\mathcal{D}_{2}(\overline{\omega}_{2d+2})=\overline{\omega}_{\kappa(d)}$.

###### Proof.

Let ${\bf x}_{0},{\bf x}_{1},\ldots,{\bf x}_{d}$ be vertices of a regular
$d$-simplex on $S^{d-1}$ such that $\overline{\omega}_{2d+2}=\\{\pm{\bf
x}_{i}:i=0,1,\ldots,d\\}$. The inclusion
$\overline{\omega}_{\kappa(d)}\subset\mathcal{D}_{2}(\overline{\omega}_{2d+2})$
holds. Indeed, choose any $\frac{d+1}{2}$-element subset
$I\subset\\{0,1,\ldots,d\\}$. If $k$ is an index in $I$, then

$\left(\frac{2\sqrt{d}}{d+1}\sum\limits_{i\in I}{\bf x}_{i}\right)\cdot{\bf
x}_{k}=\frac{2\sqrt{d}}{d+1}\left(1-\frac{1}{d}\cdot\frac{d-1}{2}\right)=\frac{1}{\sqrt{d}}.$
(18)

If $k$ is an index not in $I$, then

$\left(\frac{2\sqrt{d}}{d+1}\sum\limits_{i\in I}{\bf x}_{i}\right)\cdot{\bf
x}_{k}=\frac{2\sqrt{d}}{d+1}\cdot\left(-\frac{1}{d}\cdot\frac{d+1}{2}\right)=-\frac{1}{\sqrt{d}}.$
(19)

Then every point from $\overline{\omega}_{\kappa(d)}$ forms only two distinct
values of the dot product with points from $\overline{\omega}_{2d+2}$ (which
are $\pm\frac{1}{\sqrt{d}}$) and, hence, is contained in
$\mathcal{D}_{2}(\overline{\omega}_{2d+2})$.

To show the opposite inclusion, choose any ${\bf
y}\in\mathcal{D}_{2}(\overline{\omega}_{2d+2})$. Since points ${\bf
x}_{0},{\bf x}_{1},\ldots,{\bf x}_{d}$ form a $2$-design, the set
$\overline{\omega}_{2d+2}$ is a $3$-design. Since
$\mathcal{D}_{2}(\overline{\omega}_{2d+2})\neq\emptyset$ (it contains
$\overline{\omega}_{\kappa(d)}$), the set $\overline{\omega}_{2d+2}$ is
$2$-stiff. By Proposition 7.2, we then have ${\bf y}\cdot{\bf
x}_{i}=\pm\frac{1}{\sqrt{d}}$, $i=0,1,\ldots,d$. Since $d+1$ is even and

$\sum_{i=0}^{d}{\bf y}\cdot{\bf x}_{i}={\bf y}\cdot\sum_{i=0}^{d}{\bf
x}_{i}={\bf y}\cdot{\bf 0}=0,$

exactly half of dot products ${\bf y}\cdot{\bf x}_{i}$ equal
$\frac{1}{\sqrt{d}}$ and the other half equals $-\frac{1}{\sqrt{d}}$. Let
$I\subset\\{0,1,\ldots,d\\}$ be the $\frac{d+1}{2}$-element subset such that
${\bf y}\cdot{\bf x}_{i}=\frac{1}{\sqrt{d}}$, $i\in I$. Pick some index $m\in
I$ and some index $k\in J:=\\{0,1,\ldots,d\\}\setminus I$. Then for every
$i\in I\setminus\\{m\\}$, we have ${\bf y}\cdot({\bf x}_{i}-{\bf x}_{m})=0$
and, for every $i\in J\setminus\\{k\\}$, we have ${\bf y}\cdot({\bf
x}_{i}-{\bf x}_{k})=0$. Let us show that the set

$T:=\\{{\bf x}_{i}-{\bf x}_{m}\\}_{i\in I\setminus\\{m\\}}\cup\\{{\bf
x}_{i}-{\bf x}_{k}\\}_{i\in J\setminus\\{k\\}}$

of $d-1$ vectors in $\mathbb{R}^{d}$ (orthogonal to ${\bf y}$) is linearly
independent. Let $c_{i}$, $i\in\\{0,1,\ldots,d\\}\setminus\\{m,k\\}$ be any
numbers such that

$\sum\limits_{i\in I\setminus\\{m\\}}c_{i}({\bf x}_{i}-{\bf
x}_{m})+\sum\limits_{i\in J\setminus\\{k\\}}c_{i}({\bf x}_{i}-{\bf
x}_{k})={\bf 0}.$

Then

$\sum\limits_{i=0\atop i\neq m,k}^{d}c_{i}({\bf x}_{i}-{\bf
x}_{m})-\left(\sum\limits_{i\in J\setminus\\{k\\}}c_{i}\right)({\bf
x}_{k}-{\bf x}_{m})={\bf 0}.$ (20)

Since ${\bf x}_{0},\ldots,{\bf x}_{d}$ form a non-degenerate simplex in
$\mathbb{R}^{d}$, the set of vectors ${\bf x}_{i}-{\bf x}_{m}$, where
$i\in\\{0,1,\ldots,d\\}\setminus\\{m\\}$ is linearly independent. Then (20)
implies that $c_{i}=0$, $i\neq m,k$; that is, $T$ is linearly independent.
Thus, ${\bf y}$ appears to be orthogonal to the $(d-1)$-dimensional subspace
spanned by $T$. Then ${\bf y}$ belongs to the intersection of the one-
dimensional subspace $T^{\bot}$ with $S^{d-1}$; that is there exist only two
(opposite) vectors that ${\bf y}$ can equal to.

Let ${\bf u}:=\frac{2\sqrt{d}}{d+1}\sum_{j\in I}{\bf x}_{j}$. Then ${\bf u}\in
S^{d-1}\cap\overline{\omega}_{\kappa(d)}$ and equation (18) implies that, for
every $i\in I\setminus\\{m\\}$, we have ${\bf u}\cdot({\bf x}_{i}-{\bf
x}_{m})=\frac{1}{\sqrt{d}}-\frac{1}{\sqrt{d}}=0$. Equation (19) implies that,
for every $i\in J\setminus\\{k\\}$, we have ${\bf u}\cdot({\bf x}_{i}-{\bf
x}_{k})=-\frac{1}{\sqrt{d}}+\frac{1}{\sqrt{d}}=0$. Thus, ${\bf u}\in
T^{\bot}\cap S^{d-1}$. This implies that ${\bf y}={\bf u}$ or ${\bf y}=-{\bf
u}$. Since $\sum_{i=0}^{d}{\bf x}_{i}={\bf 0}$, we have $-{\bf
u}=\frac{2\sqrt{d}}{d+1}\sum_{j\in J}{\bf
x}_{j}\in\overline{\omega}_{\kappa(d)}$. Thus, ${\bf
y}\in\overline{\omega}_{\kappa(d)}$. ∎

######  Lemma 8.3.

For every $d\geq 3$ odd, the configuration
$\overline{\omega}_{\kappa(d)}\subset S^{d-1}$ is a spherical $3$-design.

###### Proof.

To prove the lemma, we will use Theorem 7.1. Since
$\overline{\omega}_{\kappa(d)}$ is centrally symmetric and Gegenbauer
polynomials $P_{n}^{(d-1)}$, $n=1,3$, are odd, we have

$M_{n}^{(d-1)}(\overline{\omega}_{\kappa(d)}):=\sum\limits_{{\bf
x}\in\overline{\omega}_{\kappa(d)}}\sum\limits_{{\bf
y}\in\overline{\omega}_{\kappa(d)}}P_{n}^{(d-1)}({\bf x}\cdot{\bf y})=0$ (21)

for $n=1,3$. It remains to show equality (21) for $n=2$. Let ${\bf x}_{0},{\bf
x}_{1},\ldots,{\bf x}_{d}$ be the vertices of the regular simplex in the
definition of the configuration $\overline{\omega}_{\kappa(d)}$ in (17). Let
${\bf x}_{I}:=\frac{2\sqrt{d}}{d+1}\sum_{i\in I}{\bf x}_{i}$, where $I$ is any
$\ell:=\frac{d+1}{2}$-element subset of $\\{0,1,\ldots,d\\}$. Let $0\leq
k\leq\ell$ be any integer and $J\subset\\{0,1,\ldots,d\\}$ be any
$\ell$-element subset whose intersection with $I$ has exactly $k$ elements.
There are $\binom{\ell}{k}\binom{\ell}{\ell-k}=\binom{\ell}{k}^{2}$ such sets
$J$ and, for every such $J$,

$\begin{split}{\bf x}_{I}\cdot{\bf
x}_{J}&=\frac{4d}{(d+1)^{2}}\left(\sum_{i\in I\cap J}{\bf x}_{i}+\sum_{i\in
I\setminus J}{\bf x}_{i}\right)\cdot\sum_{j\in J}{\bf x}_{j}\\\
&=\frac{d}{\ell^{2}}\left(\sum_{i\in I\cap
J}\left(1-\frac{\ell-1}{d}\right)+\sum_{i\in I\setminus
J}\left(-\frac{\ell}{d}\right)\right)=\frac{2k}{\ell}-1,\end{split}$

$k=0,1,\ldots,\ell$. Then

$\begin{split}M_{2}^{(d-1)}&(\overline{\omega}_{\kappa(d)})=\kappa(d)\sum\limits_{k=0}^{\ell}\binom{\ell}{k}^{2}P_{2}^{(d-1)}\left(\frac{2k}{\ell}-1\right)\\\
&=\frac{d\kappa(d)}{d-1}\sum\limits_{k=0}^{\ell}\binom{\ell}{k}^{2}\left(\left(\frac{2k}{\ell}-1\right)^{2}-\frac{1}{d}\right)\\\
&=\frac{d\kappa(d)}{d-1}\left(\frac{4}{\ell^{2}}\sum\limits_{k=0}^{\ell}k^{2}\binom{\ell}{k}^{2}-\frac{4}{\ell}\sum\limits_{k=0}^{\ell}k\binom{\ell}{k}^{2}+\frac{d-1}{d}\sum\limits_{k=0}^{\ell}\binom{\ell}{k}^{2}\right).\end{split}$
(22)

Using the standard formula
$\sum\limits_{k=0}^{\ell}\binom{\ell}{k}^{2}=\binom{2\ell}{\ell}$ (which can
be seen by counting the total number of subsets $J$ above), we obtain that

$\sum\limits_{k=0}^{\ell}k^{2}\binom{\ell}{k}^{2}=\sum\limits_{k=1}^{\ell}\ell^{2}\binom{\ell-1}{k-1}^{2}=\ell^{2}\sum\limits_{k=0}^{\ell-1}\binom{\ell-1}{k}^{2}=\ell^{2}\binom{2\ell-2}{\ell-1}.$

Finally, from (21) we obtain that

$\begin{split}&0=\frac{M_{1}^{(d-1)}(\overline{\omega}_{\kappa(d)})}{\kappa(d)}=\sum\limits_{k=0}^{\ell}\binom{\ell}{k}^{2}P_{1}^{(d-1)}\left(\frac{2k}{\ell}-1\right)\\\
&=\sum\limits_{k=0}^{\ell}\binom{\ell}{k}^{2}\left(\frac{2k}{\ell}-1\right)=\frac{2}{\ell}\sum\limits_{k=0}^{\ell}k\binom{\ell}{k}^{2}-\binom{2\ell}{\ell}.\end{split}$

Then $\sum_{k=0}^{\ell}k\binom{\ell}{k}^{2}=\frac{\ell}{2}\binom{2\ell}{\ell}$
and from (22) we obtain

$M_{2}^{(d-1)}(\overline{\omega}_{\kappa(d)})=\frac{d\kappa(d)}{d-1}\left(4\binom{2\ell-2}{\ell-1}-2\binom{2\ell}{\ell}+\frac{2\ell-2}{2\ell-1}\binom{2\ell}{\ell}\right)=0.$

Thus, $\overline{\omega}_{\kappa(d)}$ is a spherical $3$-design. ∎

######  Lemma 8.4.

For every $d\geq 3$ odd,
$\mathcal{D}_{2}(\overline{\omega}_{\kappa(d)})=\overline{\omega}_{2d+2}$.

###### Proof.

Relations (18) and (19) imply that
$\overline{\omega}_{2d+2}\subset\mathcal{D}_{2}(\overline{\omega}_{\kappa(d)})$.
Let ${\bf x}$ be any element of
$\mathcal{D}_{2}(\overline{\omega}_{\kappa(d)})$ and ${\bf x}_{0},{\bf
x}_{1},\ldots,{\bf x}_{d}$ be the vertices of the regular simplex in the
definition of the configuration $\overline{\omega}_{\kappa(d)}$ in (17). There
exist numbers $\alpha_{0},\alpha_{1},\ldots,\alpha_{d}$ such that
$\sum_{i=0}^{d}\alpha_{i}=1$ and ${\bf x}=\sum_{i=0}^{d}\alpha_{i}{\bf
x}_{i}$. Let $I$ be any $(d+1)/2$-element subset of the set
$\\{0,1,\ldots,d\\}$. Then relations (18) and (19) imply that

$\begin{split}\theta:=\left(\frac{2\sqrt{d}}{d+1}\sum\limits_{i\in I}{\bf
x}_{i}\right)\cdot{\bf x}&=\left(\frac{2\sqrt{d}}{d+1}\sum\limits_{i\in I}{\bf
x}_{i}\right)\cdot\left(\sum\limits_{j\in I}\alpha_{j}{\bf
x}_{j}+\sum\limits_{j\notin I}\alpha_{j}{\bf x}_{j}\right)\\\
&=\frac{1}{\sqrt{d}}\left(\sum\limits_{j\in I}\alpha_{j}-\sum\limits_{j\notin
I}\alpha_{j}\right).\end{split}$

The set $\overline{\omega}_{\kappa(d)}$ is a $3$-design by Lemma 8.3. Since
$\mathcal{D}_{2}(\overline{\omega}_{\kappa(d)})\neq\emptyset$ (it contains
$\overline{\omega}_{2d+2}$), the set $\overline{\omega}_{\kappa(d)}$ is
$2$-stiff. Then in view of Proposition 7.2, the above dot product is
$\theta=\pm\frac{1}{\sqrt{d}}$. Then $\sum_{i\in I}\alpha_{i}-\sum_{i\notin
I}\alpha_{i}=\pm 1$. Since $\sum_{i=0}^{d}\alpha_{i}=1$, for any $I$, we have
$\sum_{i\in I}\alpha_{i}=1$ or $\sum_{i\in I}\alpha_{i}=0$.

Since ${\bf x}$ and each ${\bf x}_{i}$ lie on the unit sphere, we have

$\begin{split}\left|{\bf
x}\right|^{2}&=\left(\sum\limits_{i=0}^{d}\alpha_{i}{\bf
x}_{i}\right)^{2}=\sum\limits_{i=0}^{d}\alpha_{i}^{2}-\frac{1}{d}\sum\limits_{i\neq
j}\alpha_{i}\alpha_{j}=\left(1+\frac{1}{d}\right)\sum\limits_{i=0}^{d}\alpha_{i}^{2}-\frac{1}{d}\left(\sum\limits_{i=0}^{d}\alpha_{i}\right)^{2}\\\
&=\left(1+\frac{1}{d}\right)\sum\limits_{i=0}^{d}\alpha_{i}^{2}-\frac{1}{d}=1,\end{split}$

which implies that $\sum_{i=0}^{d}\alpha_{i}^{2}=1$.

Choose arbitrary indices $j\neq k$ and let $J\subset\\{0,1,\ldots,d\\}$ be any
$(d+1)/2$-element subset such that $k\in J$ and $j\notin J$. Let
$J_{1}:=(J\cup\\{j\\})\setminus\\{k\\}$. Since each sum $\sum_{i\in
J}\alpha_{i}$ and $\sum_{i\in J_{1}}\alpha_{i}$ equals $0$ or $1$, the
difference

$\alpha_{j}-\alpha_{k}=\sum_{i\in J_{1}}\alpha_{i}-\sum_{i\in J}\alpha_{i}$

can only equal $1$, $-1$, or $0$. Let
$\alpha:=\max\limits_{i=\overline{0,d}}\alpha_{i}$. If it were that
$\alpha_{i}=\alpha$ for all indices $i$, then $\alpha_{i}$ would equal
$\frac{1}{d+1}$ for all $i$ contradicting the equality of $\sum_{i\in
J}\alpha_{i}$ to $0$ or $1$. Thus, for some index $\ell$, we have
$\alpha_{\ell}<\alpha$. For every index $i$, $\alpha-\alpha_{i}$ can only
equal $0$ or $1$ (since $\alpha$ is some maximal $\alpha_{j}$). Then
$\alpha_{i}=\alpha$ or $\alpha_{i}=\alpha-1$ for every index $i$. Let $n$ be
the number of indices $i$ such that $\alpha_{i}=\alpha$. Since the sums of
$\alpha_{i}$’s and of their squares both equal $1$, we obtain the system of
equations

$\begin{cases}n\alpha+(d+1-n)(\alpha-1)=1,\cr
n\alpha^{2}+(d+1-n)(\alpha-1)^{2}=1.\cr\end{cases}$

Multiplying the first equation by $\alpha$ and subtracting it from the second
equation, we obtain that $(d+1-n)(1-\alpha)=1-\alpha$. This yields two
solutions: $\alpha=1$, $n=1$ and $\alpha=\frac{2}{d+1}$, $n=d$. In the first
case, we have $\alpha_{\ell}=1$ for some index $\ell$ and $\alpha_{i}=0$ for
$i\neq\ell$. Then ${\bf x}={\bf x}_{\ell}\in\overline{\omega}_{2d+2}$. In the
second case, we have $\alpha_{\ell}=\frac{1-d}{d+1}$ for some index $\ell$ and
$\alpha_{i}=\frac{2}{d+1}$, $i\neq\ell$. Then ${\bf
x}=\frac{2}{d+1}\sum\limits_{i:i\neq\ell}{\bf x}_{i}+\frac{1-d}{d+1}{\bf
x_{\ell}}$. Using the equality $\sum_{i=0}^{d}{\bf x}_{i}={\bf 0}$, we obtain
that ${\bf x}=-{\bf x}_{\ell}\in\overline{\omega}_{2d+2}$. Thus,
$\mathcal{D}_{2}(\overline{\omega}_{\kappa(d)})\subset\overline{\omega}_{2d+2}$
and the assertion of the lemma follows. ∎

We are ready to prove the main result of this section.

######  Theorem 8.5.

Let $d\geq 3$ be odd and $f:[0,4]\to(-\infty,\infty]$ be a function continuous
on $(0,4]$ with $f(0)=\lim\limits_{t\to 0^{+}}f(t)$ and a convex derivative
$f^{\prime\prime}$ on $(0,4)$. Then

  * (i)

the potential $p_{f}(\cdot,\overline{\omega}_{2d+2})$ of the symmetrized
regular $d$-simplex $\overline{\omega}_{2d+2}$ attains its absolute minimum
over $S^{d-1}$ at every point of the set $\overline{\omega}_{\kappa(d)}$;

  * (ii)

the potential $p_{f}(\cdot,\overline{\omega}_{\kappa(d)})$ of the
configuration $\overline{\omega}_{\kappa(d)}$ attains its absolute minimum
over $S^{d-1}$ at every point of the set $\overline{\omega}_{2d+2}$.

If the convexity of $f^{\prime\prime}$ is strict on $(0,4)$, then the
potential $p_{f}(\cdot,\overline{\omega}_{2d+2})$ has no other points of
absolute minimum on $S^{d-1}$ in (i) and the potential
$p_{f}(\cdot,\overline{\omega}_{\kappa(d)})$ has no other points of absolute
minimum on $S^{d-1}$ in (ii).

###### Proof of Theorem 8.5.

The set $\overline{\omega}_{2d+2}$ is a $3$-design as a symmetrization about
the origin of a $2$-design. In view of Lemma 8.2, we have
$\mathcal{D}_{2}(\overline{\omega}_{2d+2})=\overline{\omega}_{\kappa(d)}\neq\emptyset$;
that is, $\overline{\omega}_{2d+2}$ is $2$-stiff. Theorem 4.3 now implies that
the potential $p_{f}(\ \\!\cdot\ \\!,\overline{\omega}_{2d+2})$ attains its
absolute minimum on $S^{d-1}$ at every point of
$\overline{\omega}_{\kappa(d)}$.

Lemma 8.3 implies that $\overline{\omega}_{\kappa(d)}$ is a spherical
$3$-design, while Lemma 8.4 implies that
$\mathcal{D}_{2}(\overline{\omega}_{\kappa(d)})=\overline{\omega}_{2d+2}\neq\emptyset$;
that is, $\overline{\omega}_{\kappa(d)}$ is $2$-stiff. Theorem 4.3 now implies
that the potential $p_{f}(\ \\!\cdot\ \\!,\overline{\omega}_{\kappa(d)})$
attains its absolute minimum on $S^{d-1}$ at every point of
$\overline{\omega}_{2d+2}$.

If the convexity of $f^{\prime\prime}$ is strict on $(0,4)$, then by Theorem
4.3, the set
$\mathcal{D}_{2}(\overline{\omega}_{2d+2})=\overline{\omega}_{\kappa(d)}$
contains all points of absolute minimum of $p_{f}(\ \\!\cdot\
\\!,\overline{\omega}_{2d+2})$ on $S^{d-1}$, while the set
$\mathcal{D}_{2}(\overline{\omega}_{\kappa(d)})=\overline{\omega}_{2d+2}$
contains all points of absolute minimum of $p_{f}(\ \\!\cdot\
\\!,\overline{\omega}_{\kappa(d)})$ on $S^{d-1}$. ∎

##  9 A pair of stiff configurations on $S^{5}$

In this section, we consider the symmetrized Schläffi configuration
$\overline{\omega}_{54}$ and the (normalized) root system of the $E_{6}$ root
lattice, $\overline{\omega}_{72}$. Recall that the Schläffi configuration,
denoted by $\overline{\omega}_{27}$, consists of the following $N=27$ points
(see, e.g., [12]): the point $(0,0,0,0,0,1)$, ten points whose sixth
coordinate is $-1/2$, one of the first five coordinates is $\pm\sqrt{3}/2$,
and the remaining four coordinates equal $0$, and sixteen points whose sixth
coordinate is $1/4$ and each of the remaining five coordinates is
$\pm\sqrt{3}/4$ with an even number of minus signs. The set
$\overline{\omega}_{27}$ is a spherical $4$-design. The symmetrized Schläffi
configuration is then defined as
$\overline{\omega}_{54}:=\overline{\omega}_{27}\cup(-\overline{\omega}_{27})$,
which is a spherical $5$-design.

Denote by $\overline{\omega}_{72}$ the configuration consisting of $40$
vectors on $S^{5}$ with two of the first five coordinates equal
$\frac{1}{\sqrt{2}}$ or $-\frac{1}{\sqrt{2}}$ and the remaining four
coordinates equal zero and $32$ vectors of the form
$\left(\pm\frac{1}{2\sqrt{2}},\pm\frac{1}{2\sqrt{2}},\pm\frac{1}{2\sqrt{2}},\pm\frac{1}{2\sqrt{2}},\pm\frac{1}{2\sqrt{2}},\pm\frac{\sqrt{3}}{2\sqrt{2}}\right)$
with an odd number of minus signs. This is a spherical $5$-design. To show
that both configurations are $3$-stiff, we use the following two auxiliary
statements.

######  Lemma 9.1.

$\mathcal{D}_{3}(\overline{\omega}_{54})=\overline{\omega}_{72}$.

###### Proof.

One can verify directly that
$\overline{\omega}_{72}\subset\mathcal{D}_{3}(\overline{\omega}_{54})$. The
set $\overline{\omega}_{54}$ is a $5$-design; that is,
$\overline{\omega}_{54}$ is $3$-stiff. Let now ${\bf x}=(x_{1},\ldots,x_{6})$
be any vector in $\mathcal{D}_{3}(\overline{\omega}_{54})$. By Proposition
7.2, we have $D({\bf
x},\overline{\omega}_{54})=\left\\{\pm\sqrt{\frac{{3}}{8}},0\right\\}$. If
${\bf u}=(0,0,0,0,0,1)\in\overline{\omega}_{54}$, then $x_{6}={\bf x}\cdot{\bf
u}=\pm\sqrt{\frac{3}{8}}$ or $0$. For every $1\leq i\leq 5$, choose the vector
${\bf v}_{i}\in\overline{\omega}_{54}$ whose $i$-th coordinate equals
$\frac{\sqrt{3}}{2}$ and the sixth coordinate equals $-\frac{1}{2}$ (all other
coordinates are zero). Then

${\bf x}\cdot{\bf
v}_{i}=\frac{\sqrt{3}}{2}x_{i}-\frac{x_{6}}{2}=\pm\sqrt{\frac{3}{8}}\ \
\text{or}\ \ 0.$ (23)

If $x_{6}=0$, then $x_{i}=\pm\frac{1}{\sqrt{2}}$ or $0$ for $i=1,\ldots,5$.
Since $\left|{\bf x}\right|=1$, exactly two coordinates out of first five
equal $\pm\frac{1}{\sqrt{2}}$ and four remaining coordinates are zero; that
is, ${\bf x}\in\overline{\omega}_{72}$.

Assume now that $x_{6}=\pm\sqrt{\frac{3}{8}}$. Then, for each $1\leq i\leq 5$,
from (23), we have $\frac{\sqrt{3}}{2}x_{i}=\pm\frac{3}{2}\sqrt{\frac{3}{8}}$
or $\pm\frac{1}{2}\sqrt{\frac{3}{8}}$. Then $x_{i}=\pm\frac{1}{\sqrt{8}}$,
$i=1,\ldots,5$ (the values $x_{i}=\pm\frac{3}{\sqrt{8}}$ are impossible, since
$\left|x_{i}\right|\leq 1$). Therefore, ${\bf
x}=\left(\pm\frac{1}{\sqrt{8}},\pm\frac{1}{\sqrt{8}},\pm\frac{1}{\sqrt{8}},\pm\frac{1}{\sqrt{8}},\pm\frac{1}{\sqrt{8}},\pm\sqrt{\frac{3}{8}}\right)$.
Let ${\bf
w}=\left(\pm\frac{\sqrt{3}}{4},\pm\frac{\sqrt{3}}{4},\pm\frac{\sqrt{3}}{4},\pm\frac{\sqrt{3}}{4},\pm\frac{\sqrt{3}}{4},\frac{1}{4}\right)$
with an even number of minus signs. Assume to the contrary that ${\bf x}$ has
an even number of negative coordinates. Then, for each $i=1,\ldots,6$, the
product of the $i$-th coordinates of vectors ${\bf x}$ and ${\bf w}$ will be
$\pm\frac{\sqrt{3}}{4\sqrt{8}}$ and the number of negative products will be
even. This implies that possible values of dot product ${\bf x}\cdot{\bf w}$
will be $\pm\frac{3\sqrt{3}}{2\sqrt{8}}$ or $\pm\frac{\sqrt{3}}{2\sqrt{8}}$,
which are not in $D({\bf x},\overline{\omega}_{54})$. Then ${\bf x}$ must have
an odd number of negative coordinates, which makes it an element of
$\overline{\omega}_{72}$. ∎

######  Lemma 9.2.

$\mathcal{D}_{3}(\overline{\omega}_{72})=\overline{\omega}_{54}.$

###### Proof.

One can verify directly that
$\overline{\omega}_{54}\subset\mathcal{D}_{3}(\overline{\omega}_{72})$ and
that $\overline{\omega}_{72}$ is a $5$-design. Then $\overline{\omega}_{72}$
is $3$-stiff. Let ${\bf x}=(x_{1},\ldots,x_{6})$ be any point in
$\mathcal{D}_{3}(\overline{\omega}_{72})$. Assume to the contrary that the set
$S:=\\{\left|x_{i}\right|:i=1,\ldots,5\\}$ has at least two distinct non-zero
elements. Then we have $0<\left|x_{k}\right|<\left|x_{\ell}\right|$ for some
indices $1\leq k\neq\ell\leq 5$. Let ${\bf u}\in\overline{\omega}_{72}$ be one
of the four vectors with the $k$-th and $\ell$-th coordinates being
$\pm\frac{1}{\sqrt{2}}$ and all other coordinates being zero. Then dot product
${\bf x}\cdot{\bf u}$ will have four distinct values
$\frac{\pm\left|x_{k}\right|\pm\left|x_{\ell}\right|}{\sqrt{2}}$ contradicting
the assumption that ${\bf x}$ is in $\mathcal{D}_{3}(\overline{\omega}_{72})$.
Thus, $x_{1}=\ldots=x_{5}=0$ or all non-zero elements of $S$ are equal. If
$x_{1}=\ldots=x_{5}=0$, then ${\bf x}=(0,0,0,0,0,\pm
1)\in\overline{\omega}_{54}$.

Assume that all non-zero elements of $S$ are equal and denote by $a$ their
common value. Denote also $b:=\left|x_{6}\right|$. Let $G$ be the set of $32$
vectors
$\left(\pm\frac{1}{2\sqrt{2}},\pm\frac{1}{2\sqrt{2}},\pm\frac{1}{2\sqrt{2}},\pm\frac{1}{2\sqrt{2}},\pm\frac{1}{2\sqrt{2}},\pm\frac{\sqrt{3}}{2\sqrt{2}}\right)$
with an odd number of minus signs.

Assume to the contrary that $b=0$. Let $1\leq k\leq 5$ be such that
$\left|x_{i}\right|=a$ for exactly $k$ indices $i$ between $1$ and $5$
inclusive. Then $a=1/\sqrt{k}$ and the remaining $6-k$ coordinates of ${\bf
x}$ are zero. Since $\overline{\omega}_{72}$ is $3$-stiff, by Proposition 7.2,
all dot products of ${\bf x}$ with vectors from $G$ must be in the set
$\left\\{\pm\frac{\sqrt{3}}{2\sqrt{2}},0\right\\}$, in particular, there is
only one positive dot product. There is a vector ${\bf u}\in G$ with
components corresponding to non-zero $x_{i}$ having the same sign as $x_{i}$.
Then ${\bf x}\cdot{\bf u}=\frac{ka}{2\sqrt{2}}=\frac{\sqrt{k}}{2\sqrt{2}}$,
which must equal $\frac{\sqrt{3}}{2\sqrt{2}}$. Consequently, $k=3$. But then
there is a vector ${\bf v}\in G$ (with all but one components corresponding to
non-zero $x_{i}$ having the same sign as $x_{i}$) such that ${\bf x}\cdot{\bf
v}=\frac{a}{2\sqrt{2}}=\frac{1}{2\sqrt{6}}$, which is not a possible dot
product. Thus, $b$ must be positive.

If $\left|x_{i}\right|=a$ for exactly one index $1\leq i\leq 5$, then ${\bf
x}$, in particular, forms dot products $0$ and $\pm\frac{a}{\sqrt{2}}$ with
points from $\overline{\omega}_{72}$. Since $\overline{\omega}_{72}$ is
$3$-stiff, by Proposition 7.2, all these dot products must be in the set
$\left\\{\pm\frac{\sqrt{3}}{2\sqrt{2}},0\right\\}$. This implies that
$\left|x_{i}\right|=a=\frac{\sqrt{3}}{2}$ and, since $\left|{\bf x}\right|=1$,
we have $x_{6}=\pm 1/2$. Since all other coordinates of ${\bf x}$ are zero in
this case, we have ${\bf x}\in\overline{\omega}_{54}$.

Assume that $\left|x_{i}\right|=a$ for exactly $k$ indices $i$ between $1$ and
$5$ inclusive, where $2\leq k\leq 4$. Then ${\bf x}$ has at least dot products
$0,\pm\frac{a}{\sqrt{2}},\pm\sqrt{2}a$ with vectors from
$\overline{\omega}_{72}$ contradicting the assumption that ${\bf
x}\in\mathcal{D}_{3}(\overline{\omega}_{72})$.

Finally, assume that $\left|x_{1}\right|=\ldots=\left|x_{5}\right|=a$. Let
${\bf w}$ be any of the four vectors from $\overline{\omega}_{72}$ whose first
and second coordinates equal $\pm\frac{1}{\sqrt{2}}$ and the remaining four
coordinates equal $0$. Then ${\bf x}\cdot{\bf w}$ has values $\pm\sqrt{2}a$
and $0$. Then $\sqrt{2}a=\frac{\sqrt{3}}{2\sqrt{2}}$ or
$a=\frac{\sqrt{3}}{4}$. Since $\left|{\bf x}\right|=1$, we have $b=1/4$.

Assume to the contrary that $x_{i}<0$ for an odd number of indices $1\leq
i\leq 6$. Let $H_{1}$ be the set of all such indices. For any vector ${\bf
u}=(u_{1},\ldots,u_{6})\in G$, let $H_{2}:=\\{i:u_{i}<0\\}$ (by definition of
the set $G$, $\\#H_{2}$ is also odd). For every $i$, we have
$\left|x_{i}u_{i}\right|=\frac{\sqrt{3}}{8\sqrt{2}}$. Then
$x_{i}u_{i}=-\frac{\sqrt{3}}{8\sqrt{2}}$ for
$\\#H_{1}+\\#H_{2}-2(\\#(H_{1}\cap H_{2}))$ indices $i$, which is an even
number. As ${\bf u}$ runs through the set $G$, dot product ${\bf x}\cdot{\bf
u}$ can only assume a value from the set
$\left\\{\pm\frac{3\sqrt{3}}{4\sqrt{2}},\pm\frac{\sqrt{3}}{4\sqrt{2}}\right\\}$,
none of which is in the set $\left\\{\pm\frac{\sqrt{3}}{2\sqrt{2}},0\right\\}$
contradicting Proposition 7.2. Therefore, coordinates of vector ${\bf x}$ must
have an even number of minus signs, which puts vector ${\bf x}$ in
$\overline{\omega}_{54}$. Thus,
$\mathcal{D}_{3}(\overline{\omega}_{72})\subset\overline{\omega}_{54}$. ∎

We are now ready to prove the main result of this section.

######  Theorem 9.3.

Let $f:[0,4]\to(-\infty,\infty]$ be a function continuous on $(0,4]$ with
$f(0)=\lim\limits_{t\to 0^{+}}f(t)$ and a convex derivative $f^{(4)}$ on
$(0,4)$. Then

  * (i)

the potential $p_{f}(\ \\!\cdot\ \\!,\overline{\omega}_{54})$ attains its
absolute minimum over $S^{5}$ at every point of $\overline{\omega}_{72}$;

  * (ii)

the potential $p_{f}(\ \\!\cdot\ \\!,\overline{\omega}_{72})$ attains its
absolute minimum over $S^{5}$ at every point of $\overline{\omega}_{54}$.

If the convexity of $f^{(4)}$ is strict on $(0,4)$, then the potential
$p_{f}(\cdot,\overline{\omega}_{54})$ has no other points of absolute minimum
in (i) and the potential $p_{f}(\ \\!\cdot\ \\!,\overline{\omega}_{72})$ has
no other points of absolute minimum in (ii).

###### Proof of Theorem 9.3.

The set $\overline{\omega}_{54}$ is a spherical $5$-design. In view of Lemma
9.1, we have $\mathcal{D}_{3}(\overline{\omega}_{54})=\overline{\omega}_{72}$;
that is, $\overline{\omega}_{54}$ is $3$-stiff. Theorem 4.3 now implies that
the potential $p_{f}(\ \\!\cdot\ \\!,\overline{\omega}_{54})$ attains its
absolute minimum on $S^{5}$ at every point of $\overline{\omega}_{72}$. The
set $\overline{\omega}_{72}$ is a spherical $5$-design. Lemma 9.2 implies that
$\mathcal{D}_{3}(\overline{\omega}_{72})=\overline{\omega}_{54}$; that is,
$\overline{\omega}_{72}$ is $3$-stiff. Theorem 4.3 now implies that the
potential $p_{f}(\ \\!\cdot\ \\!,\overline{\omega}_{72})$ attains its absolute
minimum on $S^{5}$ at every point of $\overline{\omega}_{54}$.

If the convexity of $f^{(4)}$ is strict on $(0,4)$, then by Theorem 4.3, the
set $\mathcal{D}_{3}(\overline{\omega}_{54})=\overline{\omega}_{72}$ contains
all points of absolute minimum of $p_{f}(\ \\!\cdot\
\\!,\overline{\omega}_{54})$ on $S^{5}$, while the set
$\mathcal{D}_{3}(\overline{\omega}_{72})=\overline{\omega}_{54}$ contains all
points of absolute minimum of $p_{f}(\ \\!\cdot\ \\!,\overline{\omega}_{72})$
on $S^{5}$. ∎

##  10 A pair of stiff configurations on $S^{6}$

On $S^{6}$, the kissing arrangement of $N=56$ points, denoted by
$\overline{\omega}_{56}$, was proved in [11] to be $3$-stiff. We find that its
dual is the set of $N=126$ minimal non-zero vectors of $E_{7}$ root lattice,
denoted by $\overline{\omega}_{126}$, which is also $3$-stiff. The
configuration $\overline{\omega}_{56}$ can be defined as the set of points on
the intersection of the sphere $S^{7}\subset\mathbb{R}^{8}$ with the seven-
dimensional linear subspace $\mathcal{H}$ of $\mathbb{R}^{8}$ perpendicular to
vector $(1,1,1,1,1,1,1,1)$ that have two coordinates equal to
$-\frac{3}{2\sqrt{6}}$ and six coordinates equal to $\frac{1}{2\sqrt{6}}$ or
two coordinates equal to $\frac{3}{2\sqrt{6}}$ and six coordinates equal to
$-\frac{1}{2\sqrt{6}}$. There are only three distinct values of the dot
product between any two distinct vectors in $\overline{\omega}_{56}$, which
are $\pm\frac{1}{3}$ and $-1$. It is a known $3$-sharp configuration.

The configuration $\overline{\omega}_{126}$ can be described as the
arrangement of $126$ points on the intersection $S^{7}\cap\mathcal{H}$ in
$\mathbb{R}^{8}$ that have one coordinate equal to $\frac{1}{\sqrt{2}}$, one
coordinate equal to $-\frac{1}{\sqrt{2}}$, and six zero coordinates or four
coordinates equal to $\frac{1}{2\sqrt{2}}$ and four coordinates equal to
$-\frac{1}{2\sqrt{2}}$. One can also think of $\overline{\omega}_{126}$ as the
set of minimal non-zero vectors of $E_{8}$ root lattice whose coordinates sum
to zero.

To prove the main result of this section, we use the following two auxiliary
statements.

######  Lemma 10.1.

$\mathcal{D}_{3}(\overline{\omega}_{56})=\overline{\omega}_{126}.$

###### Proof.

It is not difficult to verify directly that
$\overline{\omega}_{126}\subset\mathcal{D}_{3}(\overline{\omega}_{56})$. To
show that $\mathcal{D}_{3}(\overline{\omega}_{56})$ is contained in
$\overline{\omega}_{126}$, we will consider these configurations as subsets of
the sphere

$\widetilde{S}^{6}:=S^{7}\cap\mathcal{H}\subset\mathbb{R}^{8}.$

Let ${\bf x}=(x_{1},\ldots,x_{8})\in\widetilde{S}^{6}$ be arbitrary point in
$\mathcal{D}_{3}(\overline{\omega}_{56})$. For every pair of indices $1\leq
i\neq j\leq 8$, let ${\bf u}$ be the vector in $\overline{\omega}_{56}$ with
the $i$-th and the $j$-th coordinates equal to $-\frac{3}{2\sqrt{6}}$. Since
$x_{1}+\ldots+x_{8}=0$, we have

${\bf x}\cdot{\bf u}=\frac{1}{2\sqrt{6}}\left(-3x_{i}-3x_{j}+\sum_{k:k\neq
i,j}x_{k}\right)=-\frac{2}{\sqrt{6}}(x_{i}+x_{j})$

and ${\bf x}\cdot(-{\bf u})=\frac{2}{\sqrt{6}}(x_{i}+x_{j})$. Since
$\overline{\omega}_{56}$ is a $5$-design and
$\mathcal{D}_{3}(\overline{\omega}_{56})\neq\emptyset$, the set
$\overline{\omega}_{56}$ is $3$-stiff. Since ${\bf
x}\in\mathcal{D}_{3}(\overline{\omega}_{56})$, by Proposition 7.2, we have

$V:=\\{\pm(x_{i}+x_{j}):1\leq i\neq j\leq
8\\}\subset\left\\{\pm\frac{1}{\sqrt{2}},0\right\\}.$ (24)

Assume that ${\bf x}$ has at least three distinct coordinates, say
$x_{k}<x_{\ell}<x_{m}$. Then $V$ contains the numbers
$x_{k}+x_{\ell}<x_{k}+x_{m}<x_{\ell}+x_{m}$. This forces
$x_{k}+x_{\ell}=-\frac{1}{\sqrt{2}}$, $x_{k}+x_{m}=0$, and
$x_{\ell}+x_{m}=\frac{1}{\sqrt{2}}$. The only solution to this system is
$x_{k}=-\frac{1}{\sqrt{2}}$, $x_{\ell}=0$, and $x_{m}=\frac{1}{\sqrt{2}}$. If
for some $i\neq k,\ell,m$, we had $x_{i}\neq 0$, then $V$ would contain the
fourth element: $x_{i}+x_{m}>\frac{1}{\sqrt{2}}$ if $x_{i}>0$ or
$x_{i}+x_{k}<-\frac{1}{\sqrt{2}}$ if $x_{i}<0$. Thus,
$x_{k}=-\frac{1}{\sqrt{2}}$, $x_{m}=\frac{1}{\sqrt{2}}$, and the remaining six
coordinates are $0$; i.e., ${\bf x}\in\overline{\omega}_{126}$.

Assume now that ${\bf x}$ has at most two distinct coordinates. If all
coordinates of ${\bf x}$ were the same, then ${\bf x}$ would have to be a zero
vector, since the sum of its coordinates is zero. Thus, ${\bf x}$ has exactly
two distinct coordinates. Assume to the contrary that one of the two values of
coordinates of ${\bf x}$ appears only once. Denote this value by $c$ and let
the other value equal $b$. Then $c=-7b\neq 0$ and $V$ contains four distinct
values: $\pm 6b,\pm 2b$. This contradicts (24).

Thus, each of the two values of coordinates of ${\bf x}$ appears more than
once. Denote them so that $b<c$. Then $V$ contains the numbers $2b<b+c<2c$,
which forces $2c=\frac{1}{\sqrt{2}}$ and $2b=-\frac{1}{\sqrt{2}}$; that is,
all coordinates of ${\bf x}$ are $\pm\frac{1}{2\sqrt{2}}$. Since the sum of
coordinates of ${\bf x}$ is zero, four of its coordinates equal
$\frac{1}{2\sqrt{2}}$ and four equal $-\frac{1}{2\sqrt{2}}$; i.e., ${\bf
x}\in\overline{\omega}_{126}$. ∎

######  Lemma 10.2.

$\mathcal{D}_{3}(\overline{\omega}_{126})=\overline{\omega}_{56}$.

###### Proof.

It is not difficult to verify directly that
$\overline{\omega}_{56}\subset\mathcal{D}_{3}(\overline{\omega}_{126})$. To
show that $\mathcal{D}_{3}(\overline{\omega}_{126})$ is contained in
$\overline{\omega}_{56}$, we will consider these configurations as subsets of
the sphere $\widetilde{S}^{6}$. Let ${\bf x}\in\widetilde{S}^{6}$ be an
arbitrary point in $\mathcal{D}_{3}(\overline{\omega}_{126})$. For any $1\leq
i\neq j\leq 8$, let ${\bf u}\in\overline{\omega}_{126}$ be the vector whose
$i$-th coordinate is $\frac{1}{\sqrt{2}}$, $j$-th coordinate equals
$-\frac{1}{\sqrt{2}}$, and the remaining six coordinates equal zero. Then
${\bf x}\cdot{\bf u}=\frac{x_{i}-x_{j}}{\sqrt{2}}$.

Assume to the contrary that vector ${\bf x}$ has at least three distinct
coordinates, which we denote by $x_{k}<x_{\ell}<x_{m}$. Then ${\bf x}\cdot{\bf
u}$ has, in particular, four distinct values
$\pm\frac{x_{\ell}-x_{k}}{\sqrt{2}}$ and $\pm\frac{x_{m}-x_{k}}{\sqrt{2}}$
contradicting the fact that ${\bf
x}\in\mathcal{D}_{3}(\overline{\omega}_{126})$. Thus, there are at most two
distinct numbers among coordinates of ${\bf x}.$ If all coordinates of ${\bf
x}$ were equal, since they sum to zero, ${\bf x}$ would have to be a zero
vector contradicting the assumption that ${\bf x}\in\widetilde{S}^{6}$.

Thus, there are exactly two distinct numbers among coordinates of ${\bf x}$,
which we denote by $b<c$. By choosing vector ${\bf
u}\in\overline{\omega}_{126}$ with $\frac{1}{\sqrt{2}}$ and
$-\frac{1}{\sqrt{2}}$ on appropriate positions and $0$ on the remaining six
positions, we get value ${\bf x}\cdot{\bf u}=\frac{c-b}{\sqrt{2}}$. Since
${\bf x}\in\mathcal{D}_{3}(\overline{\omega}_{126})$, by Proposition 7.2,
${\bf x}\cdot{\bf u}\in\\{\pm\frac{1}{\sqrt{3}},0\\}$. The only value the
positive dot product $\frac{c-b}{\sqrt{2}}$ can have is $\frac{1}{\sqrt{3}}$;
that is, $c=b+\sqrt{\frac{2}{3}}$. Denote by $k$ the number of coordinates of
${\bf x}$ that equal $c$. Then $(8-k)b+k\left(b+\sqrt{\frac{2}{3}}\right)=0$
and $(8-k)b^{2}+k\left(b+\sqrt{\frac{2}{3}}\right)^{2}=1$. From the first
equation, we have $b=-\frac{k}{8}\sqrt{\frac{2}{3}}$. Then the second equation
implies that $k(8-k)=12$; that is, $k=2$ or $6$. When $k=2$, we have
$b=-\frac{1}{4}\sqrt{\frac{2}{3}}=-\frac{1}{2\sqrt{6}}$ and
$c=\frac{3}{4}\sqrt{\frac{2}{3}}=\frac{3}{2\sqrt{6}}$. When $k=6$, we have
$b=-\frac{3}{4}\sqrt{\frac{2}{3}}=-\frac{3}{2\sqrt{6}}$ and
$c=\frac{1}{4}\sqrt{\frac{2}{3}}=\frac{1}{2\sqrt{6}}$. In both cases, ${\bf
x}\in\overline{\omega}_{56}$. ∎

We are now ready to prove the main result of this section.

######  Theorem 10.3.

Let $f:[0,4]\to(-\infty,\infty]$ be a function continuous on $(0,4]$ with
$f(0)=\lim\limits_{t\to 0^{+}}f(t)$ and a convex derivative $f^{(4)}$ on
$(0,4)$. Then

  * (i)

the potential $p_{f}(\ \\!\cdot\ \\!,\overline{\omega}_{56})$ attains its
absolute minimum over $S^{6}$ at every point of $\overline{\omega}_{126}$;

  * (ii)

the potential $p_{f}(\ \\!\cdot\ \\!,\overline{\omega}_{126})$ attains its
absolute minimum over $S^{6}$ at every point of $\overline{\omega}_{56}$.

If the convexity of $f^{(4)}$ is strict on $(0,4)$, then the potential
$p_{f}(\cdot,\overline{\omega}_{56})$ has no other points of absolute minimum
in (i) and the potential $p_{f}(\cdot,\overline{\omega}_{126})$ has no other
points of absolute minimum in (ii).

We remark that one universal minimum of $\overline{\omega}_{56}$ and the value
of the absolute minimum of its potential on $S^{6}$ were found in [11].

###### Proof of Theorem 10.3.

Since $\overline{\omega}_{56}$ is a spherical $5$-design and, by Lemma 10.1,
$\mathcal{D}_{3}(\overline{\omega}_{56})=\overline{\omega}_{126}\neq\emptyset$,
the configuration $\overline{\omega}_{56}$ is $3$-stiff (see also [11]). Then
we apply Theorem 4.3 to establish item (i). The set $\overline{\omega}_{126}$
is a spherical $5$-design. Lemma 10.2 implies that
$\mathcal{D}_{3}(\overline{\omega}_{126})=\overline{\omega}_{56}\neq\emptyset$;
that is, $\overline{\omega}_{126}$ is $3$-stiff. Theorem 4.3 now implies that
the potential $p_{f}(\ \\!\cdot\ \\!,\overline{\omega}_{126})$ attains its
absolute minimum on $S^{6}$ at every point of $\overline{\omega}_{56}$.

If, in addition, $f^{(4)}$ is strictly convex on $(0,4)$, Theorem 4.3 implies
that the potential $p_{f}(\cdot,\overline{\omega}_{56})$ attains its absolute
minimum over $S^{6}$ only at points of $\overline{\omega}_{126}$ and the set
$\overline{\omega}_{56}$ contains all points of absolute minimum of $p_{f}(\
\\!\cdot\ \\!,\overline{\omega}_{126})$ on $S^{6}$. ∎

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