Log-optimal configurations on the sphere

TL;DR

This paper studies optimal arrangements of points on spheres minimizing logarithmic energy, characterizes stationary configurations for specific cases, and proposes a conjecture supported by auxiliary results.

Contribution

It provides a characterization of stationary configurations for N=d+2 points on spheres and introduces new log-optimal configurations for specific point counts and dimensions.

Findings

Characterization theorem for stationary configurations when N=d+2

New log-optimal configurations for 6 points on S^3 and 7 points on S^4

A conjecture on log-optimal configurations for N=d+2 points

Abstract

In this article we consider the distribution of $N$ points on the unit sphere $\mathbb{S}^{d-1}$ in $\mathbb{R}^d$ interacting via logarithmic potential. A characterization theorem of the stationary configurations is derived when $N=d+2$ and two new log-optimal configurations minimizing the logarithmic energy are obtained for six points on $\mathbb{S}^3$ and seven points on $\mathbb{S}^4$. A conjecture on the log-optimal configurations of $d+2$ points on $\mathbb{S}^{d-1}$ is stated and three auxiliary results supporting the conjecture are presented.