Riesz Polarization Inequalities in Higher Dimensions

TL;DR

This paper establishes bounds and asymptotics for Riesz polarization quantities on general sets in higher dimensions, focusing on spheres and balls, and extends known results with new proofs and conjectures.

Contribution

It introduces new bounds and asymptotic formulas for Riesz polarization in higher dimensions, combining potential theory with elementary averaging, and provides discrete versions and independent proofs of existing results.

Findings

Derived bounds and asymptotics for Riesz polarization quantities.

Extended results to higher dimensions and general sets.

Provided new proofs and conjectures related to Riesz polarization.

Abstract

We derive bounds and asymptotics for the maximum Riesz polarization quantity $$M_n^p(A) := \max_{{\bold x}_1, {\bold x}_2, \ldots, {\bold x}_n \in A} {\min_{{\bold x} \in A}{\sum_{j=1}^n{\frac{1}{|{\bold x} - {\bold x}_j|^{p}}}}}$$ (which is $n$ times the Chebyshev constant) for quite general sets $A \subset {\Bbb R}^m$ with special focus on the unit sphere and unit ball. We combine elementary averaging arguments with potential theoretic tools to formulate and prove our results. We also give a discrete version of the recent result of Hardin, Kendall, and Saff which solves the Riesz polarization problem for the case when $A$ is the unit circle and $p>0,$ as well as provide an independent proof of their result for $p=4$ that exploits classical polynomial inequalities and yields new estimates. Furthermore, we raise some challenging conjectures.