Riesz Polarization Inequalities in Higher Dimensions

Abstract

We derive bounds and asymptotics for the maximum Riesz polarization quantity $$M_n^p(A):=\underset{{\mathbf{x}}_1,{\mathbf{x}}_2,\dots ,{\mathbf{x}}_n\in A}{\max }\underset{\mathbf{x}\in A}{\min }\sum _{j=1}^n\frac{1}{|\mathbf{x}-{\mathbf{x}}_j{|}^p}$$ (which is $n$ times the Chebyshev constant) for quite general sets $A\subset {\mathbb{R}}^m$ 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.