Polarization problem on a higher-dimensional sphere for a simplex

Abstract

We study the problem of maximizing the minimal value over the sphere $S^{d-1}\subset \mathbb R^d$ of the potential generated by a configuration of $d+1$ points on $S^{d-1}$ (the maximal discrete polarization problem). The points interact via the potential given by a function $f$ of the Euclidean distance squared, where $f:[0,4]\to (-\infty,\infty]$ is continuous (in the extended sense) and decreasing on $[0,4]$ and finite and convex on $(0,4]$ with a concave or convex derivative $f'$. We prove that the configuration of the vertices of a regular $d$-simplex inscribed in $S^{d-1}$ is optimal. This result is new for $d>3$ (certain special cases for $d=2$ and $d=3$ are also new). As a byproduct, we find a simpler proof for the known optimal covering property of the vertices of a regular $d$-simplex inscribed in $S^{d-1}$.