A Minimum Principle for Potentials with Application to Chebyshev Constants

For “Riesz-like” kernels K(x,y) = f(|x?y|) on A×A, where A is a compact d-regular set (Asubset mathbb {R}^{p}), we prove a minimum principle for potentials (U_{K}^{mu }=int K(x,y)textup {d}mu (x)), where μ is a Borel measure supported on A. Setting (P_{K}(mu )=inf _{yin A}U^{mu }(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 )→P _K (ν) as (Nto 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.