Absolute Minima of Potentials of a Certain Class of Spherical Designs

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.