A minimum principle for potentials with application to Chebyshev constants

TL;DR

This paper establishes a minimum principle for Riesz-like potentials on compact sets and applies it to connect solutions of discrete and continuous Chebyshev polarization problems.

Contribution

It introduces a minimum principle for potentials with Riesz-like kernels and demonstrates convergence of discrete solutions to continuous solutions in Chebyshev problems.

Findings

Proves a minimum principle for Riesz-like kernels on regular sets.

Shows weak-star convergence of discrete solutions to continuous solutions.

Links discrete Chebyshev problems to their continuous counterparts.

Abstract

For "Riesz-like" kernels $K(x,y)=f(|x-y|)$ on $A\times A$, where $A$ is a compact $d$-regular set $A\subset \mathbb{R}^p$, we prove a minimum principle for potentials $U_K^\mu=\int K(x,y)d\mu(x)$, where $\mu$ is a Borel measure supported on $A$. Setting $P_K(\mu)=\inf_{y\in A}U^\mu(y)$, the $K$-polarization of $\mu$, the principle is used to show that if $\{\nu_N\}$ is a sequence of measures on $A$ that converges in the weak-star sense to the measure $\nu$, then $P_K(\nu_N)\to P_K(\nu)$ as $N\to \infty$. The continuous Chebyshev (polarization) problem concerns maximizing $P_K(\mu)$ over all probability measures $\mu$ supported on $A$, while the $N$-point discrete Chebyshev problem maximizes $P_K(\mu)$ only over normalized counting measures for $N$-point multisets on $A$. We prove for such kernels and sets $A$, that if $\{\nu_N\}$ is a sequence of $N$-point measures solving the discrete problem, then every weak-star limit measure of $\nu_N$ as $N \to \infty$ is a solution to the continuous problem.