Energy optimization for distributions on the sphere and improvement to the Welch bounds

TL;DR

This paper introduces a new framework using eccentricity tensors to analyze energy minimization on the sphere, providing new proofs and improvements to Welch bounds and characterizing optimal measures with phase transition insights.

Contribution

It develops a novel approach with eccentricity tensors for energy minimization, offering new proofs and enhancements of Welch bounds and characterizations of optimal measures.

Findings

Proves energy minimization for rotationally invariant measures.

Provides improved bounds over classical Welch bounds.

Explains phase transitions in energy optimizer families.

Abstract

For any Borel probability measure on $\mathbb{R}^n$, we may define a family of eccentricity tensors. This new notion, together with a tensorization trick, allows us to prove an energy minimization property for rotationally invariant probability measures. We use this theory to give a new proof of the Welch bounds, and to improve upon them for collections of real vectors. In addition, we are able to give elementary proofs for two theorems characterizing probability measures optimizing one-parameter families of energy integrals on the sphere. We are also able to explain why a phase transition occurs for optimizers of these two families.