Absolute Minima of Potentials of a Certain Class of Spherical Designs

TL;DR

This paper uses linear programming to identify absolute minimum potential configurations on spheres, revealing that these minima are characterized by specific geometric properties and are independent of the potential function.

Contribution

It introduces a novel method to determine universal minimal configurations on spheres for a class of potentials, extending previous understanding of spherical designs.

Findings

Minimum points form exactly m distinct dot products with the configuration

Minimum configurations are independent of the potential function f

Identifies universal minima for six specific configurations on higher-dimensional spheres

Abstract

We use linear programming techniques to find points of absolute minimum over the unit sphere $S^{d}$ in $\mathbb R^{d+1}$ of the total potential of a point configuration $\omega_N\subset S^{d}$ which is a spherical $(2m-1)$-design contained in the union of some $m$ parallel hyperplanes. The interaction between points is described by the kernel $K({\bf x},{\bf y})=f(\left|{\bf x}-{\bf y}\right|^2)$, where $\left|\ \!\cdot\ \!\right|$ is the Euclidean norm in $\mathbb R^{d+1}$. The potential function $f$ is assumed to have a convex derivative $f^{(2m-2)}$. Points of minimum do not depend on $f$ and are those and only those which form exactly $m$ distinct dot products with points of $\omega_N$. The proof of this theorem was presented at a workshop at ESI in January 2022. Using this result, we find sets of universal minima of certain six configurations on higher-dimensional spheres.