Riesz polarization inequalities in higher dimensions

Abstract

We derive bounds and asymptotics for the maximum Riesz polarization quantity Mnp(A)≔maxx1,x2,…,xn∈Aminx∈A∑j=1n1∣x−xj∣p (which is n times the Chebyshev constant) for quite general sets A⊂Rm 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.