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

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 $\OL\omega_N$.