Bounds on energy and potentials of discrete measures on the sphere

TL;DR

This paper derives universal bounds for potentials of spherical designs and explores energy minimization problems, including applications to optimal charge distributions and p-frame energy on the sphere.

Contribution

It establishes universal bounds for potentials of weighted spherical designs and characterizes the classes attaining these bounds, advancing understanding of energy distribution on spheres.

Findings

Universal bounds depend only on quadrature nodes and weights.

Characterization of spherical designs that attain the bounds.

Application to optimal charge distribution and p-frame energy.

Abstract

We establish upper and lower universal bounds for potentials of weighted designs on the sphere $\mathbb{S}^{n-1}$ that depend only on quadrature nodes and weights derived from the design structure. Our bounds hold for a large class of potentials that includes absolutely monotone functions. The classes of spherical designs attaining these bounds are characterized. Additionally, we study the problem of constrained energy minimization for Borel probability measures on $\mathbb{S}^{n-1}$ and apply it to optimal distribution of charge supported at a given number of points on the sphere. In particular, our results apply to $p$-frame energy.