Covering and separation of Chebyshev points for non-integrable Riesz potentials

TL;DR

This paper studies the geometric distribution and covering properties of optimal point configurations on compact sets for Riesz potentials, revealing asymptotic behaviors and introducing weak separation concepts.

Contribution

It establishes the optimal covering order for Riesz $s$-polarization configurations when $s>d$, and analyzes their asymptotics and separation properties.

Findings

Configurations have optimal covering order for $s>d$.

Asymptotic behavior of polarization constants as $s o \infty$.

Proves weak separation for certain $s$ ranges on spheres.

Abstract

For Riesz $s$-potentials $K(x,y)=|x-y|^{-s}$, $s>0$, we investigate separation and covering properties of $N$-point configurations $\omega^*_N=\{x_1, \ldots, x_N\}$ on a $d$-dimensional compact set $A\subset \mathbb{R}^\ell$ for which the minimum of $\sum_{j=1}^N K(x, x_j)$ is maximal. Such configurations are called $N$-point optimal Riesz $s$-polarization (or Chebyshev) configurations. For a large class of $d$-dimensional sets $A$ we show that for $s>d$ the configurations $\omega^*_N$ have the optimal order of covering. Furthermore, for these sets we investigate the asymptotics as $N\to \infty$ of the best covering constant. For these purposes we compare best-covering configurations with optimal Riesz $s$-polarization configurations and determine the $s$-th root asymptotic behavior (as $s\to \infty$) of the maximal $s$-polarization constants. In addition, we introduce the notion of "weak separation" for point configurations and prove this property for optimal Riesz $s$-polarization configurations on $A$ for $s>\text{dim}(A)$, and for $d-1\leqslant s < d$ on the sphere $\mathbb{S}^d$.

Equations

$$P_s(A;{\omega }_N):=\underset{y\in A}{\inf }\sum _{j=1}^N\frac{1}{|y-x_j{|}^s}.$$

$${\mathcal{P}}_s(A;N):=\underset{{\omega }_N\subset A}{\sup }{P}_s(A;{\omega }_N).$$

$${\mathcal{P}}_s(A;N)=P_s(A;{\omega }_N^{\ast })=\sum _{j=1}^N\frac{1}{|y^{\ast }-x_j^{\ast }{|}^s}.$$

$$\underset{N\to \infty }{\lim }\frac{{\mathcal{P}}_s(A;N)}{N^{s/d}}=\frac{{\sigma }_{s,d}}{{\mathcal{H}}_d(A{)}^{s/d}}.$$

$${\mu }_N^{\ast }:=\frac{1}{N}\sum _{x\in {\omega }_N^{\ast }}{\delta }_x\stackrel{\ast }{\to }\mu ,$$

$$\mu (B)=\frac{{\mathcal{H}}_d(B\cap A)}{{\mathcal{H}}_d(A)}.$$

$$\delta ({\omega }_N):=\underset{i\ne j}{\min }|x_i-x_j|$$

$${\rho }_A({\omega }_N):=\underset{y\in A}{\max }\underset{x\in {\omega }_N}{\min }|y-x|.$$

$${\rho }_A(N):=\underset{{\omega }_N\subset A}{\min }{\rho }_A({\omega }_N),$$

$$\underset{s\to \infty }{\lim }{\left(\frac{{\mathcal{P}}_s(A;N)}{N^{s/d}}\right)}^{1/s}=\frac{1}{N^{1/d}{\rho }_A(N)}.$$

$$\underset{s\to \infty }{\lim }{\left(\frac{{\sigma }_{s,d}}{{\mathcal{H}}_d(A{)}^{s/d}}\right)}^{1/s}=\underset{s\to \infty }{\lim }\underset{N\to \infty }{\lim }{\left(\frac{{\mathcal{P}}_s(A;N)}{N^{s/d}}\right)}^{1/s}=\frac{1}{{\lim }_{N\to \infty }{N}^{1/d}{\rho }_A(N)}.$$

$$\underset{N\to \infty }{\lim }{N}^{1/2}{\rho }_A(N)=\frac{\sqrt{2}}{\sqrt[4]{27}}{\mathcal{H}}_2(A{)}^{1/2};$$

$$\underset{s\to \infty }{\lim }{\sigma }_{s,2}^{1/s}=\frac{\sqrt[4]{27}}{\sqrt{2}}.$$

$$c_1{r}^d⩽\mu (A\cap B(x,r))⩽c_2{r}^d,$$

$${\mathcal{P}}_s(A;N)⩽C_s{N}^{s/d}.$$

$$\sum _{j=1}^N\frac{1}{|y^{\ast }-x_j{|}^s}=\underset{y\in A}{\inf }\sum _{j=1}^N\frac{1}{|y-x_j{|}^s},$$

$${\mathbb{B}}^d:=\{x\in {\mathbb{R}}^d:|x|⩽1\}$$

$$\{(x_1,\dots ,x_d):x_1^2/a_1^2+\cdots +x_d^2/a_d^2⩽1\};$$

$$\delta ({\omega }_N^{\ast })=0,{\rho }_A({\omega }_N^{\ast })=1,\forall N.$$

$$x\mapsto |x-y{|}^{-s}$$

$$\#\big{(}\omega\cap B(z,\eta\cdot(\#\omega)^{-1/d})\Big{)}\leqslant M.$$

$$\delta (\omega )⩽C\cdot (\#\omega {)}^{-1/d}.$$

$$\delta (\omega )⩾\eta \cdot (\#\omega {)}^{-1/d};$$

$${\rho }_A({\omega }_N)=\underset{y\in A}{\max }\underset{x\in {\omega }_N}{\min }|y-x|⩽R_s{N}^{-1/d},$$

$$R_s:={\left(\frac{7^s\cdot C_d\cdot M\cdot s}{5^s\cdot p_s\cdot (s-d)\cdot {\eta }^d}\right)}^{\frac{1}{s-d}},$$

$$\underset{y\in A}{\inf }\sum _{x\in {\omega }_N}\frac{1}{|y-x{|}^s}⩾p_s{N}^{s/d}.$$

$$\underset{s\to \infty }{\lim }{\left(\frac{{\sigma }_{s,d}}{{\mathcal{H}}_d(A{)}^{s/d}}\right)}^{1/s}=\underset{s\to \infty }{\lim }{\left(\underset{N\to \infty }{\lim }\frac{{\mathcal{P}}_s(A;N)}{N^{s/d}}\right)}^{1/s}=\frac{1}{{\lim }_{N\to \infty }{N}^{1/d}{\rho }_A(N)}.$$

$$\underset{s\to \infty }{\lim }{\sigma }_{s,d}^{1/s}=\frac{1}{{\lim }_{N\to \infty }{N}^{1/d}{\rho }_{[0,1{]}^d}(N)}={\left(\frac{V_d}{{\Gamma }_d}\right)}^{1/d},$$

$$\underset{s\to \infty }{\lim }\underset{N\to \infty }{\lim }{N}^{1/d}{\rho }_A({\omega }_N^s)=\underset{N\to \infty }{\lim }{N}^{1/d}{\rho }_A(N).$$

$$(Mx,x):=(Mx)\cdot x,x\in {\mathbb{R}}^d.$$

Full text

The research of A. Reznikov and E. B. Saff was supported, in part, by the National Science Foundation grants DMS-1516400

The research of A. Volberg was supported, in part, by the National Science Foundation grant DMS-1600065

1. Introduction

Suppose $A$ is a compact subset of a Euclidean space $\mathbb{R}^{\ell}$ and $\omega_{N}=\{x_{1},\ldots,x_{N}\}\subset A$ is a multiset (or an $N$-point configuration); i.e., a set of points with possible repetitions and cardinality $\#\omega_{N}=N$, counting multiplicities. For a positive number $s$ we put

[TABLE]

Then the $N$-th s-polarization (or Chebyshev) constant of $A$ is defined by

[TABLE]

We note that since $A$ is compact, there exists for each $N\in\mathbb{N}$ a configuration $\omega_{N}^{*}=\{x_{1}^{*},\ldots,x_{N}^{*}\}$ and a point $y^{*}$ such that

[TABLE]

We call $\omega_{N}^{*}$ an optimal (or extremal) Riesz $s$-polarization configuration or simply an optimal configuration.

From an applications prospective, the maximal polarization problem, say on a compact surface (or body), can be viewed as the problem of determining the smallest number of sources (injectors) of a substance together with their optimal locations that can provide a required saturation of the substance at every point of the surface (body).

The general notion of polarization (or Chebyshev constants) for potentials was likely first introduced by Ohtsuka [17]. Further investigations of the asymptotic behavior as $N\to\infty$ of polarization constants as well as the asymptotic behavior of optimal configurations appear, for example, in [1], [8], [10], [9], [3], [19], [2], [4], [18].

The following result is a special case of a theorem due to Borodachov, Hardin, Reznikov and Saff [4] (see also [2]). It describes the asymptotic behavior of optimal configurations for the case of non-integrable Riesz kernels on $A$. Here and throughout we denote by $\mathcal{H}_{d}$ the Hausdorff measure on $\mathbb{R}^{\ell}$, $d\leqslant\ell$, normalized by $\mathcal{H}_{d}([0,1]^{d})=1$.

Theorem 1.1**.**

Suppose $A$ is a compact $C^{1}$-smooth $d$-dimensional manifold, embedded in $\mathbb{R}^{\ell}$ with $d\leqslant\ell$, and $\mathcal{H}_{d}(\partial A)=0$, where $\partial A$ denotes the boundary of $A$. If $s>d$, then there exists a positive finite constant $\sigma_{s,d}$ that does not depend on $A$ such that

[TABLE]

Moreover, if $\{\omega_{N}^{*}\}_{N=1}^{\infty}$ is any sequence of optimal configurations satisfying (1), then the normalized counting measures $\mu_{N}^{*}$ for the multisets $\omega_{N}^{*}$ satisfy

[TABLE]

where $\stackrel{{\scriptstyle*}}{{\to}}$ denotes convergence in the weak∗ topology, and $\mu$ is the uniform measure on $A$; i.e., for any Borel set $B\subset\mathbb{R}^{\ell}$

[TABLE]

In other words, in the limit, optimal polarization configurations $\omega_{N}^{*}$ for non-integrable Riesz potentials are uniformly distributed in the weak∗ sense. In this paper we study more distributional properties of optimal configurations $\omega_{N}^{*}$. In particular, we investigate their separation, their covering (or mesh) radius, and their connection to the ‘‘best covering problem’’ for the set $A$.

Definition 1.2**.**

Let $A$ be a compact subset of a Euclidean space. For any $N$-point configuration $\omega_{N}\subset A$, the separation constant of $\omega_{N}$ is defined by

[TABLE]

and the covering radius of $\omega_{N}$ is defined by

[TABLE]

The best $N$-point covering radius for $A$ $\rho_{A}(N)$ is given by

[TABLE]

where the minimum is taken over all $N$-point configurations $\omega_{N}\subset A$.

In approximation theory (for example, in interpolation by splines), the separation constant $\delta(\omega_{N})$ often measures ‘‘stability’’ of approximation, while the covering radius $\rho_{A}(\omega_{N})$ is involved in bounds for the error of the approximation (see, e.g., [5]). Quasi-uniform sequences; i.e., sequences $\{\omega_{N}\}_{N=2}^{\infty}$ for which the ratios $\rho_{A}(\omega_{N})/\delta(\omega_{N})$ are bounded from above, appear, for example, in a number of applications involving approximation by radial basis functions, see, e.g., [16]. Thus they play an important role in the complexity analysis for such applications.

Regarding the asymptotic behavior of polarization constants as $s$ grows large, it is known, see [2], that for a fixed $N$ we have

[TABLE]

However, the proof in [2] does not guarantee that this limit is uniform in $N$; thus it does not imply any asymptotic behavior of the constants $\sigma_{s,d}$ in (2) as $s\to\infty$. One of our main results, Theorem 2.8, shows that for a large class of $d$-dimensional sets $A$,

[TABLE]

In the case when $A\subset\mathbb{R}^{2}$ is a compact set with $\mathcal{H}_{2}(A)>0$, it is known [13] that

[TABLE]

thus from (5),

[TABLE]

For higher dimensions we prove that all limits in (5) exist.

We shall work primarily with the class of $d$-regular sets.

Definition 1.3**.**

A compact set $A\subset\mathbb{R}^{\ell}$ is called $d$-regular if there exist a measure $\mu$ supported on $A$ and two positive constants $c_{1}$ and $c_{2}$ such that for any $x\in A$ and any positive $r<\operatorname{diam}(A)$, we have

[TABLE]

where $B(x,r)$ is the open ball in $\mathbb{R}^{\ell}$ with center $x$ and radius $r$.

The following estimate from above for $\mathcal{P}_{s}(A;N)$, which follows from [8, Theorem 2.4] and its proof, will be useful for our investigation.

Theorem 1.4**.**

If $A\subset\mathbb{R}^{\ell}$, $\ell\geqslant d$, $\mathcal{H}_{d}(A)>0$ and $s>d$, then there exists a constant $C_{s}>0$, that depends on $d$, $A$ and $s$ such that, for any positive integer $N$,

[TABLE]

Moreover, $C_{s}$ can be chosen so that there exists a constant $C_{0}$ with the property that for large values of $s$ we have $1\leqslant(C_{s})^{1/s}\leqslant C_{0}$.

The following immediate consequence of this theorem will be proved in Section 8.

Proposition 1.5**.**

With the hypotheses of Theorem 1.4, let $\omega_{N}=\{x_{j}\}_{j=1}^{N}$ be a fixed $N$-point configuration on $A$. There exists a positive constant $c_{s}$, independent of $N$ and $\omega_{N}$, with the following property: if $y^{*}=y^{*}_{s}\in A$ is a point such that

[TABLE]

then $|y^{*}-x_{j}|\geqslant c_{s}N^{-1/d}$ for each $j=1,\ldots,N$. Moreover, $c_{s}$ can be chosen so that $\lim_{s\to\infty}c_{s}^{1/s}=1$.

Furthermore, the same is true for $s\in[d-1,d)$ when $A=\mathbb{S}^{d}$, the $d$-dimensional unit sphere in $\mathbb{R}^{d+1}$.

We next introduce the main class of sets $A$ that we will consider.

Definition 1.6**.**

A compact set $A\subset\mathbb{R}^{d}$ is called a body if $A\not=\emptyset$ and $A=\textup{Clos}(\textup{Int}(A))$. We say that a body $A\subset\mathbb{R}^{d}$ is strongly convex if it is convex and its boundary $\partial A$ is a $(d-1)$-dimensional $C^{2}$-smooth manifold with non-degenerate Gaussian curvature ***Such conditions appear in many problems in harmonic analysis, see, e.g., [12]..

This class includes the closed unit ball

[TABLE]

and ellipsoids

[TABLE]

however, it does not include cubes and polyhedra.

The paper is organized as follows. In Section 2 we state and discuss our main results. In Section 3 we prove a ‘weak separation’ result for strongly convex bodies. In Section 4 we prove the ‘weak separation’ for the unit cube $[0,1]^{d}$, and in Section 5 we prove it for the unit sphere $\mathbb{S}^{d}$ and spherical caps in $\mathbb{S}^{d}$. Further, in Section 6, we derive a criterion for a sequence of configurations to have an optimal order of covering radius $\rho_{A}(\omega_{N})$. We also show that configurations $\omega^{*}_{N}$ that are optimal for $\mathcal{P}_{s}(A;N)$ satisfy this criterion if $A$ is strongly convex, a cube, a sphere, or a spherical cap. And, in Section 7, we connect the asymptotic behavior of the constant $\sigma_{s,d}$ as $s\to\infty$ with the asymptotic behavior of the best covering radius $\rho_{N}(A)$, where $A$ is any of the sets just mentioned. We prove Proposition 1.5 in Section 8 and in the Appendix (Section 9) we present equivalent definitions of best covering for the space $\mathbb{R}^{d}$.

2. Main results

For strongly convex bodies $A\subset\mathbb{R}^{d}$ the separation and covering properties of extremal configurations $\omega_{N}^{*}$ for $\mathcal{P}_{s}(A;N)$, in general, depend on the parameter $s$. Here we shall prove ‘weak separation’ and covering properties for $s>d$. In contrast, it is known [8] that for the closed $d$-dimensional unit ball $\mathbb{B}^{d}\subset\mathbb{R}^{d}$ and for $0<s\leqslant d-2$, the unique optimal $N$-point $s$-polarization configuration $\omega^{*}_{N}$ is $\omega^{*}_{N}=\{0,\ldots,0\}$; thus,

[TABLE]

The main reason behind this is that the function

[TABLE]

is superharmonic when $s\leqslant d-2$.

Our first goal is to establish for the non-integrable case $s>d$ a weak-separation property in the following sense.

Definition 2.1**.**

A family $\Omega$ of multisets $\omega$ from $A$, where $A\subset\mathbb{R}^{\ell}$ has Hausdorff dimension $d$, is called weakly well-separated with parameter $\eta>0$ if there exists an $M\in\mathbb{N}$ such that for every $\omega\in\Omega$ and every point $z\in\mathbb{R}^{\ell}$, we have

[TABLE]

It is easy to see that for a $d$-regular set $A$ there exists a positive constant $C$ such that for any configuration $\omega\subset A$ we have

[TABLE]

If for some $\eta>0$ inequality (8) holds with $M=1$ for every $\omega\in\Omega$, then

[TABLE]

therefore, we get the optimal order of separation with respect to the cardinality of $\omega$.

Definition 2.2**.**

A set $A$ is called $d$-admissible if $A\subset\mathbb{R}^{d}$ is strongly convex, or $A=\mathbb{S}^{d}\subset\mathbb{R}^{d+1}$, or $A\subset\mathbb{S}^{d}$ is a spherical cap.

We prove the following theorems.

Theorem 2.3**.**

If $d\in\mathbb{N}$, $s>d$, and the set $A$ is $d$-admissible, then there exists an $\eta>0$ such that the family $\Omega=\Omega_{s}:=\{\omega\colon P_{s}(A;\omega)=\mathcal{P}_{s}(A;\#\omega)\}$ is weakly well-separated with parameter $\eta$ and $M=2d-1$. Moreover, $\eta=\eta_{s}$ can be chosen so that $\lim_{s\to\infty}\eta_{s}^{1/s}=1$.

The same is true for $s\in[d-1,d)$ when $A=\mathbb{S}^{d}$.

The result for strongly convex bodies is proved in Section 3, while the results for the sphere and spherical caps are proved in Section 5.

Remark**.**

If $d=1$ and $A=[0,1]$, then for every $s>1$, the family $\Omega=\Omega_{s}$ is weakly well-separated with some $\eta>0$ and $M=1$.

As a consequence of the proof of Theorem 2.3, we obtain the following.

Corollary 2.4**.**

Assume $A\subset\mathbb{R}^{d}$ is a compact set and $s>d$. For every $r>0$, there exists an $\eta>0$ that depends on $r$ with the following property: if for some $z\in A$ we have $B(z,r)\subset A$, then $\#(\omega^{*}_{N}\cap B(z,\eta N^{-1/d}))\leqslant 2d-1$, where $\omega_{N}^{*}$ is optimal for $\mathcal{P}_{s}(A;N)$.

Remark**.**

As we shall show in Lemma 3.1, if $A$ is strongly convex then no points from $\omega_{N}^{*}$ can lie on the boundary $\partial A$; moreover, the distance from any point in $\omega_{N}^{*}$ to $\partial A$ is at least of the order $N^{-2/d}$.

The next theorem deals with the unit cube. For this case, our methods impose a stronger condition on the Riesz parameter $s$.

Theorem 2.5**.**

If $[0,1]^{d}\subset\mathbb{R}^{d}$, $d\geqslant 2$, denotes the unit cube and $s>3d-4$, then there exists a $\eta>0$ such that the family $\Omega=\Omega_{s}=\{\omega\colon P_{s}(A;\omega)=\mathcal{P}_{s}(A;\#\omega)\}$ is weakly well-separated with parameter $\eta$ and $M=2d-1$. Moreover, $\eta=\eta_{s}$ can be chosen so that $\lim_{s\to\infty}\eta_{s}^{1/s}=1$.

Regarding the covering radius of $N$-point configurations having a weak separation property we prove the following.

Theorem 2.6**.**

Let $\ell$, $d$ and $s$ be positive integers with $\ell\geqslant d$ and $s>d$. Suppose the compact set $A\subset\mathbb{R}^{\ell}$ with $\mathcal{H}_{d}(A)>0$ is contained in some $d$-regular compact set $\tilde{A}$. If the $N$-point configuration $\omega_{N}\subset A$ is such that for some $\eta>0$ and $M\in\mathbb{N}$ we have $\#(B(z,\eta N^{-1/d})\cap\omega_{N})\leqslant M$ for all $z\in A$, then

[TABLE]

where

[TABLE]

$C_{d}$* is a positive constant that depends only on $d$ and $A$, and $p_{s}$ is any positive constant such that*

[TABLE]

From this theorem and Theorem 2.3 we deduce the following.

Corollary 2.7**.**

If the set $A$ is $d$-admissible and $s>d$, then there exists a positive constant $R_{s}$ such that for any $N$-point configuration $\omega^{*}_{N}$ that is optimal for $\mathcal{P}_{s}(A;N)$, we have $\rho_{A}(\omega^{*}_{N})\leqslant R_{s}N^{-1/d}$. Moreover, there exists a positive constant $R_{0}$ such that for large values of $s$ we have $R_{s}\leqslant R_{0}$.

The same is true if $A=[0,1]^{d}$ and $s>3d-4$.

Corollary 2.7 implies that if $A$ is an $d$-admissible set or a unit cube, then $\rho_{A}(N)\leqslant R_{s}N^{-1/d}$ for some positive constant $R_{s}$. On the other hand, it is easy to see that in this case, for some positive constant $b$, we have $\rho_{A}(N)\geqslant bN^{-1/d}$. Fine estimates on the constant $R_{s}$ for large values of $s$ result in the following theorem dealing with the asymptotic behavior of $\mathcal{P}_{s}(A;N)^{1/s}$ as $s\to\infty$.

Theorem 2.8**.**

Suppose the set $A$ is $d$-admissible or $A=[0,1]^{d}$. Then with $\sigma_{s,d}$ as defined in Theorem 1.1, the following limits exist as positive real numbers and satisfy

[TABLE]

In particular, taking $A=[0,1]^{d}$ we obtain

[TABLE]

where the constant $\Gamma_{d}$ is the optimal covering density †††The problem of finding $\Gamma_{d}$ is known in [7] as “finding the thinnest covering of $\mathbb{R}^{d}$.” of the space $\mathbb{R}^{d}$ (see [7, Chapter 2] and Section 9) and $V_{d}:=\mathcal{H}_{d}(\mathbb{B}^{d})=\pi^{d/2}/\Gamma(d/2+1)$.

We remark that $\Gamma_{1}=1$ and $\Gamma_{2}=2\pi/\sqrt{27}$.

A consequence Theorem 2.8 is that, in the limit as $s\to\infty$, the covering radius of optimal Riesz $s$-polarization configurations become asymptotically best possible.

Corollary 2.9**.**

Suppose the set $A$ is $d$-admissible or $A=[0,1]^{d}$. For every $s>3d-4$, let $\omega_{N}^{s}$ be an $N$-point configuration such that $\mathcal{P}_{s}(A;N)=P_{s}(A;\omega_{N}^{s})$. Then

[TABLE]

3. Weak separation for strongly convex bodies

In what follows, we always assume $s>d$ and $A\subset\mathbb{R}^{d}$ is a strongly convex body. By $\overline{B(x,r)}$ we denote the closure of $B(x,r)$ and $I_{d-1}$ denotes the $(d-1)\times(d-1)$ identity matrix. Furthermore, the $j$’th coordinate of a point $x\in\mathbb{R}^{d}$ will be denoted by $x(j)$; we also denote by $x^{\prime}$ the $(d-1)$-dimensional vector that consists of the first $d-1$ coordinates of $x$; thus, $x=(x^{\prime},x(d))$. By $e_{1},\ldots,e_{d}$ we denote the canonical basis in $\mathbb{R}^{d}$. If we have a $d\times d$ matrix $M$, we put

[TABLE]

To establish Theorem 2.3 we begin with two lemmas about the behavior of extremal configurations for $\mathcal{P}_{s}(A;N)$ near the boundary $\partial A$.

Lemma 3.1**.**

There exists a constant $b_{s}>0$ with the following property: for all $N\geqslant 1$, if $\omega^{*}_{N}$ is an extremal configuration for $\mathcal{P}_{s}(A;N)$ and $x\in\omega^{*}_{N}$, then $\textup{dist}(x,\partial A)>b_{s}N^{-2/d}$ . Moreover, $b_{s}$ can be chosen so that $\lim_{s\to\infty}b_{s}^{1/s}=1$.

Remark**.**

Let $x_{\partial}\in\partial A$ and make a rotation so that in the neighborhood $B(x_{\partial},r)$ the manifold $\partial A$ is given by $\{(x^{\prime},x(d))\colon x(d)=f(x^{\prime})\}$ with $\nabla f(x_{\partial}^{\prime})=0$ and the matrix $\textup{d}^{2}f(x)$ is non-positive for $x\in\partial A\cap\overline{B(x_{\partial},r)}$ (this can be done since $A$ is convex). Moreover, $r$ can be chosen sufficiently small so that

[TABLE]

We notice that the Gaussian curvature of $\partial A$ at $x_{\partial}$ is equal to the product of eigenvalues of the matrix $\textup{d}^{2}f(x_{\partial}^{\prime})$. Since in Theorem 2.3 we assume the Gaussian curvature is non-zero, the manifold $\partial A$ is compact and $C^{2}$-smooth and $\text{d}^{2}f\leqslant 0$, we deduce that there exists a constant $C_{A}>0$ such that $\text{d}^{2}f(x^{\prime})\leqslant-C_{A}I_{d-1}$ for every $x\in B(x_{\partial},r)$, where $C_{A}$ does not depend on $x_{\partial}$.

Proof of Lemma 3.1.

Take a point $x_{\partial}\in\partial A$ for which $|x-x_{\partial}|=\textup{dist}(x,\partial A)$. We can make a rotation and assume $x=x_{\partial}-cN^{-2/d}\cdot e_{d}$. We show that this is impossible if $c$ is sufficiently small.

Let $f$ be the function from the above remark. For a small positive number $\varepsilon$ consider a point

[TABLE]

and a configuration $\widetilde{\omega}_{N}:=(\omega^{*}_{N}\setminus\{x\})\cup\{\tilde{x}\}$. Consider a point $\tilde{y}$ such that

[TABLE]

Since $\omega^{*}_{N}$ is an extremal configuration, we have

[TABLE]

which after utilizing the definition of $\widetilde{\omega}_{N}$ implies

[TABLE]

Using that $\tilde{x}=x-\varepsilon e_{d}$, we get

[TABLE]

or

[TABLE]

Since $\varepsilon$ is an arbitrarily small number, we can assume $\varepsilon/2\leqslant cN^{-2/d}$. Then we obtain

[TABLE]

On the other hand, since $A$ is a convex set, and the plane $\{z\in\mathbb{R}^{d}\colon z(d)=x_{\partial}(d)\}$ is tangent to $\partial A$, we have $\tilde{y}(d)\leqslant x_{\partial}(d)$.

We now estimate the diameter of the set

[TABLE]

Since $A$ is strongly convex, we obviously have $A\cap\{z\in\mathbb{R}^{d}\colon z(d)=x_{\partial}(d)\}=\{x_{\partial}\}$. Thus, $\textup{diam}(S(N,c))\to 0$ as $c\to 0$. If $c$ is chosen small enough, then $S(N,c)\subset B(x_{\partial},\eta)\cap A$ for some $\eta>0$. Therefore, if $y$ belongs to $S(N,c)$, then for some $\xi\in B(x_{\partial},\eta)$ we have

[TABLE]

which implies

[TABLE]

thus, for a suitable constant $C_{B}$,

[TABLE]

Therefore, since $\varepsilon\leqslant 2cN^{-2/d}$,

[TABLE]

for some constant $\tau$ that does not depend on $s$. For $c$ sufficiently small, this inequality contradicts Proposition 1.5 and so the lemma follows. ∎

In the next lemma we show that if $x\in A$ is close to $\partial A$ in one direction, then its distance in orthogonal directions can be estimated from below.

Lemma 3.2**.**

Let $\omega^{*}_{N}$ be an extremal configuration for $\mathcal{P}_{s}(A;N)$ and $x\in\omega^{*}_{N}$. Assume $\tau$ is a sufficiently small positive number that does not depend on $N$. If $\textup{dist}(x,\partial A)=|x-x_{\partial}|$ with $x-x_{\partial}$ parallel to $e_{d}$, then the estimate $|x-x_{\partial}|<\tau N^{-1/d}$ implies $x\pm\tau N^{-1/d}e_{j}\in A$ for every $j=1,\ldots,d-1$.

Proof.

Again let $f$ be as in the above remark. Arguing as in the preceding lemma, we see that we need to show that $|x-x_{\partial}|<\tau N^{-1/d}$ implies $x(d)\leqslant f(x^{\prime}\pm\tau N^{-1/d}e_{j}^{\prime})$. Notice that since $x\in\omega^{*}_{N}$, we know that $|x-x_{\partial}|>cN^{-2/d}$ for some constant $c$. We apply the Taylor formula again:

[TABLE]

Since the boundary $\partial A$ is compact and smooth, we can always assume $\textup{d}^{2}f(\xi^{\prime})>-CI_{d-1}$ for some positive constant $C$. Thus,

[TABLE]

if $\tau$ is sufficiently small. ∎

We are ready to prove Theorem 2.3.

Proof of Theorem 2.3 for a strongly convex set $A$.

We argue by contradiction. Suppose there exists small number $\eta>0$ and an extremal configuration $\omega^{*}_{N}=\{x_{1},\ldots,x_{N}\}$ such that $\{x_{1},\ldots,x_{2d}\}\subset B(z,\eta N^{-1/d})$. Consider

[TABLE]

Since $\hat{x}\in B(z,\eta N^{-1/d})$, we have $|x_{j}-\hat{x}|\leqslant 2\eta N^{-1/d}$ for every $j=1,\ldots,2d$.

Fix a small number $\tau>\eta$. We will choose it later to be a multiple of $\eta$. Set $\varepsilon:=\tau N^{-1/d}$. We consider two cases.

Case 1:

$\textup{dist}(\hat{x},\partial A)\geqslant\varepsilon$. Define $2d$ points as follows:

[TABLE]

Since $\textup{dist}(\hat{x},\partial A)\geqslant\varepsilon$, these points belong to $A$. Define $\widetilde{\omega}_{N}:=\{\tilde{x}_{1},\ldots,\tilde{x}_{2d},\tilde{x}_{2d+1},\ldots,\tilde{x}_{N}\}$, where $\tilde{x}_{j}:=x_{j}$ for $j\geqslant 2d+1$. Let $\tilde{y}$ be such that

[TABLE]

We have

[TABLE]

and thus

[TABLE]

Set $f(x):=|\tilde{y}-x|^{-s}$. Then, from the Taylor formula about $\hat{x}$, we have for $x\in\{x_{1},\ldots,x_{2d}\}$

[TABLE]

for some $\xi=\xi(x)\in B(\hat{x},|x-\hat{x}|)$. From Proposition 1.5 we know that $|\tilde{y}-\tilde{x}_{1}|\geqslant c_{s}N^{-1/d}$. Without loss of generality we assume $\tau<c_{s}/2$, and so

[TABLE]

and

[TABLE]

Therefore, for every $j=1,\ldots,2d$ we have

[TABLE]

Summing these inequalities over $j$ and recalling that $x_{1}+\cdots+x_{2d}=2d\hat{x}$ yields

[TABLE]

Plugging this estimate into (19), we obtain

[TABLE]

We proceed with the Taylor formula for $f(\tilde{x}_{j})$. We first write it for $j=1$. Recall that $\tilde{x}_{1}=\hat{x}-\varepsilon e_{1}$. Since $|e_{1}|=1$, we get for some $\xi\in B(\hat{x},|\tilde{x}_{1}-\hat{x}|)=B(\hat{x},\varepsilon)$,

[TABLE]

Next we estimate the remainder term involving $\xi$. As before,

[TABLE]

This implies

[TABLE]

Using the formula (23) with $\tilde{x}_{1}$ replaced by $\tilde{x}_{j}$ we obtain an equation for $f(\tilde{x}_{j})$ which, when substituted along with (24) into (22), yields

[TABLE]

We remark that the first term in (25) is, up to a constant factor, the Laplacian, in $x$, of the function $f(x)$. Although $f(x)$ is neither convex nor concave (for some choices of $\tilde{y}$, about which we have no information), the Laplacian $\Delta f(x)$ is always positive, which plays an essential role in our argument. Indeed, the need for at least $2d$ points $\{x_{j}\}_{j=1}^{2d}$ enables the definition of $\{\tilde{x}_{j}\}_{j=1}^{2d}$ so that the leading terms in the Taylor formula vanish leaving the positive second term. This will enable us to arrive at a contradiction to (25) as we now explain.

Recalling from (20) that $|\tilde{y}-\hat{x}|\geqslant(c_{s}/2)\cdot N^{-1/d}$, we multiply (25) by $2|\tilde{y}-\hat{x}|^{s+2}$ and divide by $sN^{-2/d}$ to obtain

[TABLE]

Since $s>d$, this is impossible if $\tau$ is a suitable large multiple (depending on $s$) of $\eta$ and $\eta$ is small, and so the first assertion of Theorem 2.3 holds in this case. Observe that (26) fails if $\eta=\eta_{s}=c_{s}/s$ and $s$ is sufficiently large. Hence from Proposition 1.5 the family $\Omega_{s}$ is weakly well-separated with $M=2d-1$ and parameter $\eta_{s}$ with $\lim_{s\to\infty}\eta_{s}^{1/s}=1$.

Case 2:

$\textup{dist}(\hat{x},\partial A)<\varepsilon$. Without loss of generality, we assume $\bar{x}+\varepsilon e_{d}\not\in A$. We again take the point $x_{\partial}\in\partial A$ that achieves this distance and argue as in Lemma 3.2. We see that for any $j\leqslant 2d-2$ the points $\tilde{x}_{j}$, defined as above, lie in the set $A$. We redefine

[TABLE]

and let $\tilde{y}$ be as in (18). The Taylor expansions of the terms on the left in (19) yield the following analog of (25):

[TABLE]

and, consequently, we have the following analog of (26),

[TABLE]

Since $\tilde{y}(d)\leqslant x_{\partial}(d)$ and $\hat{x}(d)>x_{\partial}(d)-\tau N^{-1/d}$, we obtain

[TABLE]

therefore, (28) is impossible for suitably small choices of $\eta$ and $\tau$, which as in the Case 1 yields the assertion of Theorem 2.3. ∎

4. Weak separation for the cube

In this section we show how to modify the proof of Theorem 2.3 to a case when the boundary $\partial A$ is not smooth. Namely, we prove the weak well-separation result for the unit cube, Theorem 2.5.

We begin with the following lemma.

Lemma 4.1**.**

If $s>d$, $\omega^{*}_{N}$ is optimal for $\mathcal{P}_{s}([0,1]^{d};N)$, and $x\in\omega^{*}_{N}$, then there exists a constant $b_{s}$ that does not depend on $N$ such that

[TABLE]

Moreover, one can choose $b_{s}$ so that $\lim_{s\to\infty}b_{s}^{1/s}=1$.

Proof.

We proceed as in Lemma 3.1. Denote $v:=(1,\ldots,1)$ and $\tilde{x}:=x+\varepsilon v$. If for some small number $c$ we have $\max_{j=1,\ldots,d}x(j)\leqslant cN^{-1/d}$, then $\tilde{x}\in[0,1]^{d}$. Further, set $\widetilde{\omega}_{N}:=(\omega^{*}_{N}\setminus\{x\})\cup\{\tilde{x}\}$. If $\tilde{y}$ minimizes $P_{s}([0,1]^{d},\widetilde{\omega}_{N})$, then we have

[TABLE]

which implies

[TABLE]

Utilizing the definition of $v$ and taking $\varepsilon\leqslant cN^{-1/d}$, we obtain

[TABLE]

Therefore,

[TABLE]

If $c$ is small enough, this contradicts Proposition 1.5. ∎

We are ready to prove Theorem 2.5.

Weak separation for the cube.

We again argue by contradiction. Suppose for $\eta>0$ and an optimal Riesz $s$-polarization configuration $\omega^{*}_{N}=\{x_{1},\ldots,x_{N}\}$ we have $\{x_{1},\ldots,x_{2d}\}\subset B(z,\eta N^{-1/d})$. Define

[TABLE]

Since $\hat{x}\in B(z,\eta N^{-1/d})$, we have $|x_{j}-\hat{x}|\leqslant 2\eta N^{-1/d}$ for every $j=1,\ldots,2d$.

Consider a small number $\tau>\eta$. We will choose it later to be a multiple of $\eta$. Set $\varepsilon:=\tau N^{-1/d}$. We consider two cases.

Case 1:

$\textup{dist}(\hat{x},\partial[0,1]^{d})\geqslant\varepsilon$. In this case we proceed exactly as in the first case of Section 3 and get the same contradiction.

Case 2:

$\textup{dist}(\hat{x},\partial[0,1]^{d})<\varepsilon$. We notice that since $|\hat{x}-x_{j}|<2\eta N^{-1/d}$, Lemma 4.1 implies that $\hat{x}$ cannot be close to any vertex of the cube. Therefore, there exists at least one number $j$ such that $\hat{x}\pm\varepsilon e_{j}\in[0,1]^{d}$. Without loss of generality, $j=1$. We now assume that for some $j_{0}=1,\ldots,N$ we have $\hat{x}\pm\varepsilon e_{j}\in[0,1]^{d}$ for $j\leqslant j_{0}$, and $\hat{x}-\varepsilon e_{j}\not\in[0,1]^{d}$ for $j>j_{0}$. Cases when $\hat{x}+\varepsilon e_{j}\not\in[0,1]^{d}$ are treated similarly. We define

[TABLE]

$\tilde{x}_{k}:=\hat{x}+\varepsilon e_{\lfloor(k+1)/2\rfloor}$ for $k=2j_{0}+1,\ldots,2d$, and $\widetilde{\omega}_{N}:=\{\tilde{x}_{1},\ldots,\tilde{x}_{N}\}$, where $\tilde{x}_{j}:=x_{j}$ for $j>2d$. Let $\tilde{y}$ such that

[TABLE]

Similarly to (27), we get

[TABLE]

Notice that if $\tilde{y}(j)\geqslant\hat{x}(j)$, then

[TABLE]

If $\tilde{y}(j)<\hat{x}(j)$, then we estimate $\tilde{y}(j)-\hat{x}(j)\geqslant-\hat{x}(j)\geqslant-\varepsilon$. Since $j_{0}>1$, we have at most $d-1$ numbers $j$ with $j>j_{0}$. Therefore, (29) implies

[TABLE]

which for suitably chosen $\eta$ and $\tau$ gives a contradiction if $s>3d-4$. As with Theorem 2.3, it follows that $\eta=\eta_{s}$ can be taken so that $\lim_{s\to\infty}\eta_{s}^{1/s}=1$.

∎

5. Weak separation on the sphere and spherical caps

In this section we prove Theorem 1.6 when $A=\mathbb{S}^{d}$ or when $A\subset\mathbb{S}^{d}$ is a spherical cap. We proceed as in Section 3. However, computations will be different since the sphere $\mathbb{S}^{d}$ is not ‘‘flat’’. We start with the following result.

Theorem 5.1** (Weak separation on the sphere).**

Consider the unit sphere $\mathbb{S}^{d}\subset\mathbb{R}^{d+1}$, and $s>d$ of $s\in[d-1,d)$. Then there exists a number $\eta>0$ such that for any $N$, any optimal configuration $\omega^{*}_{N}$ and any point $z\in\mathbb{R}^{d+1}$, we have

[TABLE]

Moreover, for large values of $s$ we can choose $\eta=\eta_{s}$ with

[TABLE]

Proof.

Assume the theorem is false: there exists a ball $B(z,\eta N^{-1/d})$ and an optimal configuration $\omega^{*}_{N}=\{x_{1},\ldots,x_{N}\}$ such that $\{x_{1},\ldots,x_{2d}\}\subset B(z,\eta N^{-1/d})$.

Without loss of generality, we can assume $z^{\prime}=0$ and $z(d+1)>0$. Denote

[TABLE]

and

[TABLE]

Since $|x_{j}-z|<\eta N^{-1/d}$ for $j=1,\ldots,2d$, then $|x_{j}^{\prime}|=|x_{j}^{\prime}-z^{\prime}|<\eta N^{-1/d}$; thus

[TABLE]

and

[TABLE]

Therefore,

[TABLE]

which implies for $\eta$ sufficiently small

[TABLE]

We conclude that

[TABLE]

Since the problem is rotation-invariant, we can assume $\hat{x}=e_{d+1}=(0,0,\ldots,0,1)$ — the North pole of the sphere.

Fix a small number $\tau$, with $\eta<\tau<c_{s}/20$. We will choose $\tau$ at the end of the proof. Set

[TABLE]

Note that $\{e^{\prime}_{1},\ldots,e^{\prime}_{d}\}$ is the canonical orthonormal basis in $\mathbb{R}^{d}$; denote

[TABLE]

For $j=1,\ldots,2d$ set

[TABLE]

and $\tilde{x}_{j}:=x_{j}$ if $j>2d$. For $\widetilde{\omega}_{N}:=\{\tilde{x}_{1},\ldots,\tilde{x}_{N}\}$ let $\tilde{y}$ be such that

[TABLE]

As before, denote

[TABLE]

Estimates

[TABLE]

imply, after utilizing that $x_{j}=\tilde{x}_{j}$ for $j\geqslant 2d+1$, that

[TABLE]

Then from Taylor formula about $\hat{x}$ we have for $x\in\{x_{1},\ldots,x_{2d}\}$ for some $\xi=\xi(x)\in B(\hat{x},|x-\hat{x}|)$,

[TABLE]

Recall that if $x=x_{j}$, $1\leqslant j\leqslant 2d$, then $|x-\hat{x}|\leqslant\sqrt{5}\eta N^{-1/d}$. Moreover, we know from Lemma 1.5 that $|\tilde{y}-\tilde{x}_{j}|\geqslant c_{s}N^{-1/d}$. This implies

[TABLE]

and

[TABLE]

Therefore, for every $j=1,\ldots,2d$ we have

[TABLE]

Summing these inequalities over $j$ and recalling that $x_{1}^{\prime}+\cdots+x_{2d}^{\prime}=(2d)\cdot e^{\prime}=0$, we obtain

[TABLE]

From $|\tilde{y}(d+1)-1|\leqslant 2$ and $|x_{j}(d+1)-1|=1-x_{j}(d+1)\leqslant\eta^{2}N^{-2/d}$, we get

[TABLE]

Plugging this estimate in (31), we obtain

[TABLE]

We proceed with the Taylor formula for $f(\tilde{x}_{j})$ about $\hat{x}$. We first write it for $j=1$. Recall that $\tilde{x}_{1}=(\varepsilon e^{\prime}_{1},\sqrt{1-\varepsilon^{2}})$. Setting $v:=\tilde{x}_{1}-\hat{x}=(\varepsilon e^{\prime}_{1},\sqrt{1-\varepsilon^{2}}-1)$, we obtain for some

$\xi\in B(\hat{x},|\tilde{x}_{1}-\hat{x}|)\subset B(\hat{x},\sqrt{2}\varepsilon)$,

[TABLE]

We first estimate the remainder term involving $\xi$. As before,

[TABLE]

Thus,

[TABLE]

For every $j=1,\ldots,2d$ write the Taylor formula similar to (35); in view of the estimate (36), we get from(34),

[TABLE]

Using

[TABLE]

dividing by $s$ and multiplying by $|\tilde{y}-e|^{s+4}$, we obtain

[TABLE]

Let us simplify first two terms. Notice that $|\tilde{y}-\hat{x}|^{2}=2-2\tilde{y}\cdot\hat{x}=2-2\tilde{y}(d+1)$. We have:

[TABLE]

If $\tilde{y}(d+1)<0$, we use that $\sqrt{1-\varepsilon^{2}}-1\leqslant-\frac{\varepsilon^{2}}{2}$ to get

[TABLE]

If $\tilde{y}(d+1)\geqslant 0$, we use $\sqrt{1-\varepsilon^{2}}-1\geqslant-\frac{\varepsilon^{2}}{2}-\frac{\varepsilon^{4}}{8}$ to get

[TABLE]

Combining estimates (40) and (41), we get

[TABLE]

Plugging this estimate into (38) and dividing by $|\tilde{y}-\hat{x}|^{2}$, we obtain:

[TABLE]

We now recall that $\varepsilon=\tau N^{-1/d}$. Denote

[TABLE]

Then

[TABLE]

We should finally recall that $N^{-1/d}|\tilde{y}-\hat{x}|^{-1}\leqslant 2/c_{s}$. Thus, we can choose sufficiently small $\eta$ and $\tau$ such that the left-hand side of (44) is strictly positive, which is a contradiction. Finally, as in Section 3, for large values of $s$ we can choose $\eta=\eta_{s}$ with $\eta_{s}^{1/s}\to 1$ as $s\to\infty$. ∎

We proceed with the same statement for spherical caps $A\subset\mathbb{S}^{d}$. As in the case of bodies in $\mathbb{R}^{d}$, we will need to deal with the case when point $\hat{x}$ is near the boundary.

Corollary 5.2** (Weak separation on the caps).**

Consider the unit sphere $\mathbb{S}^{d}\subset\mathbb{R}^{d+1}$, and $s>d$. Let $A\subset\mathbb{S}^{d}$ be a spherical cap, $A=\{x\in\mathbb{S}^{d}\colon x(1)\geqslant t_{0}\}$. Then there exists a number $\eta>0$ such that for any $N$, any optimal configuration $\omega^{*}_{N}$ for $\mathcal{P}_{s}(A;N)$, and any point $z\in\mathbb{R}^{d+1}$ we have

[TABLE]

Moreover, for large values of $s$ we can choose $\eta=\eta_{s}$ so that

[TABLE]

Proof.

For the sake of simplicity, we prove this corollary for $d=2$. The case of general $d$ can be treated similarly. We also assume $t_{0}\geqslant 0$. The case $t_{0}<0$ is done through the same estimates.

We again argue by contradiction. Assume for some small $\eta>0$ there exists a ball $B(z,\eta N^{-1/2})$ and an extremal configuration $\omega^{*}_{N}=\{x_{1},\ldots,x_{N}\}$ such that $\{x_{1},\ldots,x_{4}\}\subset B(z,N^{-1/2})$. Set

[TABLE]

and

[TABLE]

Recall that $x\in A$ if and only if $x(1)\geqslant t_{0}$. Thus, we see that $\hat{x}^{\prime}\in A$, and, as before,

[TABLE]

Since the problem is rotation invariant, we can assume $\hat{x}=(\hat{t},0,\sqrt{1-\hat{t}^{2}})$ for some $\hat{t}\geqslant t_{0}$.

We denote

[TABLE]

Set $\varepsilon:=\tau N^{-1/2}$ and consider

[TABLE]

If $\tilde{x}_{1},\ldots,\tilde{x}_{4}\in A$, then we get the same contradiction as for the sphere $\mathbb{S}^{d}$. Thus, the only case we need to consider is when one of these points is not in $A$.

A direct computation shows that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Thus, $\tilde{x}_{1}(1)$ and $\tilde{x}_{3}(1)$ are greater or equal than $t_{0}$, and if $\tilde{x}_{2}(1)<t_{0}$ or $\tilde{x}_{4}(1)<t_{0}$, then

[TABLE]

If this is the case, we define the points $\tilde{x}_{1},\ldots,\tilde{x}_{4}$ differently; namely,

[TABLE]

We set $\tilde{x}_{j}:=x_{j}$ for $j>4$, $\widetilde{\omega}_{N}:=\{\tilde{x}_{1},\ldots,\tilde{x}_{N}\}$ and write the same Taylor formulas as before. We get

[TABLE]

Expanding $f(\tilde{x}_{j})$ about $\hat{x}$ as before, we get

[TABLE]

where the remainder terms are handled exactly as in (36).

We proceed with showing that the third term can not be a large negative number. In fact,

[TABLE]

If $\tilde{y}(3)<0$, we see that this expression is non-neagtive. Otherwise, plugging

[TABLE]

and $\tilde{y}(3)\leqslant\sqrt{1-t_{0}^{2}}$ into (48), we obtain

[TABLE]

for some non-negative constant $c$, which depends only on $t_{0}$. We finally show how to estimate the second term of (47). Without loss of generality, we can assume this term is negative, in particular, $\hat{t}\not=0$. The equality

[TABLE]

yields

[TABLE]

where again $c$ is a positive constant which depends only on $t_{0}$. On the other hand,

[TABLE]

Thus, inequality (47) implies

[TABLE]

which is impossible since $s>2$. ∎

6. Proofs of covering results

Proof of Theorem 2.6.

Fix an integer $N$. Since $\tilde{A}$ is a $d$-regular compact set, there exists a finite family of sets $\{Q_{\alpha}\}_{\alpha}$ with the following properties:

1. (i)

$\tilde{A}=\cup_{\alpha}Q_{\alpha}$ and the interiors of the sets $Q_{\alpha}$ are disjoint; furthermore, $\mu(Q_{\alpha})=0$ for every $\alpha$, where $\mu$ is the measure from Definition 1.3; 2. (ii)

There exists a positive constant $a_{1}$ that does not depend on $N$, and points $z_{\alpha}\in Q_{\alpha}$ such that $B(z_{\alpha},a_{1}\eta N^{-1/d})\cap\tilde{A}\subset Q_{\alpha}\subset B(z_{\alpha},\eta N^{-1/d})$.

For the construction of such sets see, e.g., [6]. Notice that since $Q_{\alpha}\subset B(z_{\alpha},\eta N^{-1/d})$, we have $\#(Q_{\alpha}\cap\omega_{N})\leqslant M$.

Let $\mathcal{A}$ denote the set of indices $\alpha$ such that $Q_{\alpha}\cap\omega_{N}\not=\emptyset$. Since every $Q_{\alpha}$ can contain no more than $M$ points from $\omega_{N}$, we deduce that number of such indices is at least as large as $N/M$.

Hereafter we follow an argument in [11].

Without loss of generality, we assume $\rho_{A}(\omega_{N})\geqslant 5\eta N^{-1/d}$. Let $y\in A$ be such that $\min_{x_{k}\in\omega_{N}}|y-x_{k}|=\rho_{A}(\omega_{N})$. For every $x_{j}\in\omega_{N}$ let $\alpha_{j}=\alpha$ denote the index such that $x_{j}\in Q_{\alpha}$ for some $\alpha$. If $x\in Q_{\alpha}$, then

[TABLE]

Consequently,

[TABLE]

Furthermore,

[TABLE]

which implies

[TABLE]

For each $x_{j}\in Q_{\alpha}$ we see from (49) that

[TABLE]

Since $B(z_{\alpha},a_{1}\eta N^{-1/d})\cap\tilde{A}\subset Q_{\alpha}$, we have by the $d$-regularity condition that $\mu(Q_{\alpha})\geqslant c_{1}\cdot\eta^{d}/N$, where the positive constant $c_{1}$ does not depend on $s$. This implies from assumption (12) that

[TABLE]

where $c_{2}$ does not depend on $s$. This yields, for $C_{d}:=c_{1}^{-1}\cdot c_{2}$,

[TABLE]

which implies

[TABLE]

as claimed. ∎

Proof of Corollary 2.7.

First, we prove that for any $\omega_{N}$ that is extremal for $\mathcal{P}_{s}(A;N)$, there exists a positive constant $p_{s}$ with

[TABLE]

We prove it for strongly convex $A\subset\mathbb{R}^{d}$ or $A=[0,1]^{d}$. The case $A=\mathbb{S}^{d}$ is similar. First, notice that for any $z\in A$ we have $A\subset z+[-\operatorname{diam}(A),\operatorname{diam}(A)]^{d}=:Q$. For a fixed $N$ and a fixed constant $a$, consider a maximal set $\mathcal{E}$ such that for any $x,y\in\mathcal{E}$ we have $|x-y|\geqslant aN^{-1/d}$. The maximality of $\mathcal{E}$ implies that

[TABLE]

thus $\rho_{A}(\mathcal{E})\leqslant aN^{-1/d}$.

On the other hand, we see that the sets $B(x,(a/3)N^{-1/d})\cap Q$ are disjoint. Thus,

[TABLE]

which implies

[TABLE]

where $c_{1}$ and $c_{2}$ are positive constants that depend on $d$. We now choose $a$ such that $c_{2}a^{-d}=1$. This implies that there exists an $N$-point set $\widetilde{\omega}_{N}$ such that

[TABLE]

where the number $a$ depends only on $A$ and $d$. In particular, $\rho_{A}(\widetilde{\omega}_{N})\leqslant aN^{-1/d}$.

Observe that

[TABLE]

Thus, we can apply Theorem 2.6 with $p_{s}=a^{-s}$ to obtain

[TABLE]

for

[TABLE]

where $\eta_{s}$ is the constant from Theorem 2.3 or Theorem 2.5.

To complete the proof, recall that we have $\lim_{s\to\infty}\eta_{s}^{1/s}=1$, therefore for large values of $s$ we have $R_{s}\leqslant R_{0}$ for some positive $R_{0}$. ∎

7. Proof of Best Covering Results

We begin by remarking that in Section 6 we have seen that if $A$ is $d$-regular, then for some positive constants $a$ and $b$ we have $aN^{-1/d}\leqslant\rho_{A}(N)\leqslant bN^{-1/d}$, where $\rho_{A}(N)$ is defined in (4).

Proof of Theorem 2.8.

Using the same argument as in (51), we see that

[TABLE]

Therefore,

[TABLE]

which implies

[TABLE]

On the other hand, for a fixed positive integer $N$ and large $s$ consider an $N$-point configuration $\omega^{*}_{N}=\{x_{1},\ldots,x_{N}\}$ such that $\mathcal{P}_{s}(A;N)=P_{s}(A;\omega^{*}_{N})$. Corollary 2.7 implies that if $s$ is large enough, then $\rho_{A}(\omega^{*}_{N})\leqslant R_{0}N^{-1/d}$, where $R_{0}$ depends neither on $N$, nor on $s$. We also recall that the Theorems 2.3 and 2.5 imply that for any large value of $s$ there exists a number $\eta_{s}>0$ such that for any $z\in\mathbb{R}^{d}$ we have $\#(\omega^{*}_{N}\cap B(z,\eta_{s}N^{-1/d}))\leqslant 2d-1$ and $\lim_{s\to\infty}\eta_{s}^{1/s}=1$.

We now take a point $y\in A$ such that

[TABLE]

and set

[TABLE]

where $n$ is an integer with $n\geqslant 2$. Since the open ball $B(y,\rho_{A}(\omega^{*}_{N}))$ does not intersect $\omega^{*}_{N}$, we have

[TABLE]

Notice that for any $n\geqslant 2$ we have $B_{n}\subset B(y,\,nR_{0}N^{-1/d})$; thus, there exists a constant $\tilde{C}_{1}$ that does not depend on $s$ such that the annulus $B_{n}$ can be covered by $\tilde{C}_{1}R_{0}^{d}n^{d}\eta_{s}^{-d}=:C_{2}n^{d}\eta_{s}^{-d}$ balls of radius $\eta N^{-1/d}$. Thus, for any $n\geqslant 2$ we have

[TABLE]

For $y$ defined in (53) we have

[TABLE]

By the definition of $B_{n}$, for any $x\in B_{n}$ we have $|y-x|\geqslant(n-1)\rho_{A}(\omega^{*}_{N})$, which implies

[TABLE]

Dividing by $N^{s/d}$ and using that $\rho_{A}(\omega^{*}_{N})\geqslant\rho_{A}(N)$, we obtain

[TABLE]

which implies

[TABLE]

Taking $\limsup_{s\to\infty}$, we obtain

[TABLE]

Estimates (52) and (57) imply that $\lim_{N\to\infty}N^{1/d}\rho_{A}(N)$ and $\lim_{s\to\infty}\left(\lim_{N\to\infty}\mathcal{P}_{s}(A;N)N^{-s/d}\right)^{1/s}$ exist and satisfy

[TABLE]

∎

As an immediate consequence of Theorem 2.8 we state the following corollary about behavior of covering radii of optimal $s$-Riesz polarization configurations as $s\to\infty$.

Corollary 7.1**.**

Suppose $A$ is a $d$-admissible set or $A=[0,1]^{d}$. For every $N\geqslant 1$ and every $s>d$ fix an $N$-point configuration $\omega_{N}^{s}$ such that $\mathcal{P}_{s}(A;N)=P_{s}(A;\omega^{s}_{N})$. Then the following limits exist and satisfy

[TABLE]

Proof.

Arguing as in (51), we get that

[TABLE]

which implies from (13) that

[TABLE]

On the other hand, arguing as in (54), (55) and (56) we get

[TABLE]

and (58) follows. ∎

8. Proof of Proposition 1.5

Proof of Proposition 1.5 for $s>d$.

Take a positive integer $N$, an $N$-point configuration $\omega_{N}$ and the point $y^{*}$. Theorem 1.4 implies, for any $j=1,\ldots,N$,

[TABLE]

therefore, $|y^{*}-x_{j}|\geqslant C_{s}^{-1/s}\cdot N^{-1/d}=:c_{s}N^{-1/d}$. ∎

To prove Proposition 1.5 for the case $A=\mathbb{S}^{d}$ and $s\in[d-1,d)$ we set

[TABLE]

Then it is well known (see, e.g., [15]) that if $s\in(0,d)$ then $U(y)$ is constant of $\mathbb{S}^{d}$, and we denote this constant by $\gamma_{s,d}$ ‡‡‡$\gamma_{s,d}$ is the Wiener constant (maximal $s$-energy constant) on $\mathbb{S}^{d}$..

We need the following lemma, which can be found in [14].

Lemma 8.1**.**

For each $s\in[d-1,d)$ there exists a constant $C=C(s,d)$ such that for every $y$ with $|y|=1+N^{-1/d}$ we have

[TABLE]

Furthermore, if for a constant $c$ and an $N$-point configuration $\omega_{N}\subset\mathbb{S}^{d}$ we have $U(y)\leqslant c\cdot U^{\omega_{N}}(y)$, where

[TABLE]

then the same inequality holds for every $y\in\mathbb{R}^{d+1}$.

Proof of Proposition 1.5 for $A=\mathbb{S}^{d}$ and $s\in[d-1,d)$.

Fix an $N$-point configuration $\omega_{N}=\{x_{1},\ldots,x_{N}\}$ and set $\gamma:=P_{s}(\mathbb{S}^{d};\omega_{N})$. For every $y\in\mathbb{S}^{d}$ we have

[TABLE]

thus, for every $y$ with $|y|=1+N^{-1/d}$ we have

[TABLE]

Notice that

[TABLE]

which implies that for every $y$ with $|y|=1+N^{-1/d}$, we have

[TABLE]

With $y^{*}$ as in the statement of Proposition 1.5, set $y:=(1+N^{-1/d})\cdot y^{*}$. Then for every $j=1,\ldots,N$ we have $|x_{j}-y|\geqslant|x_{j}-y^{*}|$. Therefore, for every $i=1,\ldots,N$, if follows from (61) that

[TABLE]

We now use that $|x_{i}-y|\geqslant N^{-1/d}$ to get

[TABLE]

which completes the proof. ∎

9. Appendix: equivalent definition of best covering of the Euclidean space $\mathbb{R}^{d}$

Assume $\mathcal{B}\subset\mathbb{R}^{d}$ is a family of unit balls. The density of $\mathcal{B}$ is defined by

[TABLE]

whenever the limit exists. The optimal covering density for $\mathbb{R}^{d}$ is defined by

[TABLE]

where the infimum is taken over all families $\mathcal{B}$ that cover $\mathbb{R}^{d}$.

It is known, see [7, Chapter 2] and [2], that $\Gamma_{1}$ is attained for balls centered on the lattice $2\mathbb{Z}$ and $\Gamma_{2}$ is attained for balls centered on the properly rescaled equi-triangular lattice. For higher dimensions no explicit results are known; however, if we minimize only over lattices, then it is known that for $d\leqslant 5$ an optimal lattice is the properly rescaled $A_{d}:=\{(x_{1},\ldots,x_{d+1})\in\mathbb{Z}^{d+1}\colon x_{1}+\cdots+x_{d+1}=0\}$, which is a lattice in a $d$-dimensional hyperplane.

We start by proving the following lemma.

Lemma 9.1**.**

If $V_{d}=\mathcal{H}_{d}(\mathbb{B}^{d})$, $\mathcal{B}$ covers $\mathbb{R}^{d}$ and the limit (62) exists, then

[TABLE]

Conversely, if the limit in the right-hand side exists, then $\Delta(\mathcal{B})$ exists as well and $\Delta(\mathcal{B})/V_{d}$ is equal to this limit.

Proof.

Define \mathcal{B}_{R}:=\left\{B\in\mathcal{B}\colon\mbox{center of B$is in$[-R,R]^{d}}\right\}. We estimate

[TABLE]

On the other hand, if $B\cap[-R,R]^{d}\not=\emptyset$, then the center of $B$ is in $[-R-2,R+2]^{d}$. Therefore,

[TABLE]

Estimates (63) and (64) obviously imply assertion of the lemma. ∎

We continue with more equivalent definitions of $\Gamma_{d}$. For a compact set $A\subset\mathbb{R}^{d}$ and a positive number $r$ put

[TABLE]

A simple rescaling argument yields for every $R>0$

[TABLE]

We show the following.

Theorem 9.2**.**

For every $d\in\mathbb{N}$ we have

[TABLE]

Proof.

The existence of

[TABLE]

as well as the last equality follows from Theorem 2.8. The equalities

[TABLE]

are straightforward and left to the reader. We derive the first equality in (65). For a small $\varepsilon>0$ take a set $\mathcal{B}$ such that

[TABLE]

and

[TABLE]

where $\mathcal{B}_{R}$ is defined as in preceding proof. As in the proof of Lemma 9.1, we have

[TABLE]

therefore

[TABLE]

Consequently,

[TABLE]

In view of the arbitrariness of $\varepsilon$, we get

[TABLE]

To prove the opposite inequality, we fix a large number $R_{0}$ and choose a configuration $\omega$ with $\#\omega=N_{[-R_{0},R_{0}]^{d}}(1)$ and

[TABLE]

Define

[TABLE]

then obviously

[TABLE]

Fix a number $R>R_{0}$ and choose an integer $n$ such that $(2n-1)R_{0}\leqslant R\leqslant(2n+1)R_{0}$. Then

[TABLE]

Since

[TABLE]

and

[TABLE]

we get

[TABLE]

Therefore,

[TABLE]

which implies, in view of Lemma 9.1, that

[TABLE]

From of the arbitrariness of $R_{0}$ and the estimate (66), the lemma follows. ∎