A minimum principle for potentials with application to Chebyshev constants

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 $U_K^μ=\int K(x,y)dμ(x)$, where $μ$ is a Borel measure supported on $A$. Setting $P_K(μ)=\inf_{y\in A}U^μ(y)$, the $K$-polarization of $μ$, the principle is used to show that if $\{ν_N\}$ is a sequence of measures on $A$ that converges in the weak-star sense to the measure $ν$, then $P_K(ν_N)\to P_K(ν)$ as $N\to \infty$. The continuous Chebyshev (polarization) problem concerns maximizing $P_K(μ)$ over all probability measures $μ$ supported on $A$, while the $N$-point discrete Chebyshev problem maximizes $P_K(μ)$ only over normalized counting measures for $N$-point multisets on $A$. We prove for such kernels and sets $A$, that if $\{ν_N\}$ is a sequence of $N$-point measures solving the discrete problem, then every weak-star limit measure of $ν_N$ as $N \to \infty$ is a solution to the continuous problem.