Absolute Minima of Potentials of Certain Regular Spherical Configurations

TL;DR

This paper uses approximation theory to identify absolute minima of potentials for specific spherical configurations, including designs and polyhedral vertices, revealing symmetry-based minimal arrangements for various potentials.

Contribution

It introduces a method to find absolute minima of potentials for certain spherical designs and configurations, extending to classical and monotone potentials, with explicit examples.

Findings

Minimum for icosahedron vertices at dodecahedron vertices

Minimum for dodecahedron vertices at icosahedron vertices

Minimum for E8 lattice vectors at a specific 2160-point configuration

Abstract

We use methods of approximation theory to find the absolute minima on the sphere of the potential of spherical $(2m-3)$-designs with a non-trivial index $2m$ that are contained in a union of $m$ parallel hyperplanes, $m\geq 2$, whose locations satisfy certain additional assumptions. The interaction between points is described by a function of the dot product, which has positive derivatives of orders $2m-2$, $2m-1$, and $2m$. This includes the case of the classical Coulomb, Riesz, and logarithmic potentials as well as a completely monotone potential of the distance squared. We illustrate this result by showing that the absolute minimum of the potential of the set of vertices of the icosahedron on the unit sphere $S^2$ in $\mathbb R^3$ is attained at the vertices of the dual dodecahedron and the one for the set of vertices of the dodecahedron is attained at the vertices of the dual icosahedron. The absolute minimum of the potential of the configuration of $240$ minimal vectors of $E_8$ root lattice normalized to lie on the unit sphere $S^7$ in $\mathbb R^8$ is attained at a set of $2160$ points on $S^7$ which we describe.