Positive definiteness and the Stolarsky invariance principle

TL;DR

This paper explores the relationships between positive definiteness, energy minimization, and discrepancy, establishing equivalences and extending the Stolarsky Invariance Principle to broader contexts on compact spaces.

Contribution

It demonstrates the equivalence of positive definiteness with various properties and generalizes the Stolarsky Invariance Principle to compact spaces.

Findings

Conditional positive definiteness is equivalent to convexity of the energy functional.

The paper proves a general form of the Stolarsky Invariance Principle.

Energy minimization characterizations are extended to broader settings.

Abstract

In this paper we elaborate on the interplay between energy optimization, positive definiteness, and discrepancy. In particular, assuming the existence of a $K$-invariant measure $\mu$ with full support, we show that conditional positive definiteness of a kernel $K$ is equivalent to a long list of other properties: including, among others, convexity of the energy functional, inequalities for mixed energies, and the fact that $\mu$ minimizes the energy integral in various senses. In addition, we prove a very general form of the Stolarsky Invariance Principle on compact spaces, which connects energy minimization and discrepancy and extends several previously known versions.