Energy on spheres and discreteness of minimizing measures

TL;DR

This paper investigates energy minimization on spheres, revealing that optimal measures tend to be discrete or supported on small sets, especially for certain potentials and parameters, with proofs of existence and support properties.

Contribution

It demonstrates that minimizers of specific energy functionals on spheres are discrete or supported on small sets, especially when the potential is not positive definite or for non-even integer p.

Findings

Support of minimizers has empty interior for non-even integer p.

Existence of discrete minimizers for polynomial and analytic potentials.

Optimal measures tend to be discrete or supported on small sets.

Abstract

In the present paper we study the minimization of energy integrals on the sphere with a focus on an interesting clustering phenomenon: for certain types of potentials, optimal measures are discrete or are supported on small sets. In particular, we prove that the support of any minimizer of the $p$-frame energy has empty interior whenever $p$ is not an even integer. A similar effect is also demonstrated for energies with analytic potentials which are not positive definite. In addition, we establish the existence of discrete minimizers for a large class of energies, which includes energies with polynomial potentials.