Min-Max Polarization for Certain Classes of Sharp Configurations on the Sphere

TL;DR

This paper studies optimal point configurations on spheres that minimize the maximum potential, showing sharp configurations like antipodal points or even-strength spherical designs are solutions, with maxima attained at configuration points.

Contribution

It establishes that certain sharp configurations on spheres solve the min-max potential problem, extending understanding of optimal sphere point arrangements.

Findings

Sharp configurations like antipodal points are solutions.

Maximum potential occurs at configuration points.

Results apply to potentials with specific monotonicity properties.

Abstract

We consider the problem of finding an $N$-point configuration on the sphere $S^d\subset \RR^{d+1}$ with the smallest absolute maximum value over $S^d$ of its total potential. The potential induced by each point ${\bf y}$ in a given configuration at a point ${\bf x}\in S^d$ is $f\(\left|{\bf x}-{\bf y}\right|^2\)$, where $f$ is continuous on $[0,4]$ and completely monotone on $(0,4]$, and $\left|{\bf x}-{\bf y}\right|$ is the Euclidean distance between points~${\bf x}$ and ${\bf y}$. We show that any sharp point configuration $\OL\omega_N$ on $S^d$, which is antipodal or is a spherical design of an even strength is a solution to this problem. We also prove that the absolute maximum over $S^d$ of the potential of any such configuration $\OL\omega_N$ is attained at points of $\OL\omega_N$.