Characterization of Logarithmic Fekete Critical Configurations of at Most Six Points in All Dimensions

Abstract

We consider the logarithmic Fekete problem, which consists of placing a fixed number of points on the unit sphere in $\mathbb{R}^d$, in such a way that the product of all pairs of mutual Euclidean distances is maximized or, equivalently, so that their logarithmic energy is minimized. Using tools from Computational Algebraic Geometry, we find and classify all critical configurations for this problem when considering at most six points in every dimension $d$. In particular, our approach gives new proofs of several key results appearing in the literature, with the benefit of using a unified approach. Furthermore, for seven points in $S^2$, we characterize the global minimizer among critical configurations having at least one pair of antipodal points, and give numerical evidence to support the conjecture that this configuration is also the unrestricted global minimizer.