Transfinite diameter, Chebyshev constant and energy on locally compact spaces

TL;DR

This paper explores the relationships between transfinite diameter, Chebyshev constant, and Wiener energy in potential theory, establishing conditions under which these quantities are equal and characterizing kernels with the maximum principle.

Contribution

It proves that for kernels with the maximum principle, transfinite diameter, Chebyshev constant, and Wiener energy are equal for all compact sets, and characterizes kernels satisfying the converse.

Findings

Quantitative equality of potential theoretic quantities under the maximum principle

Characterization of kernels with the maximum principle via Chebyshev constant and transfinite diameter

Examples demonstrating the sharpness of the theoretical results

Abstract

We study the relationship between transfinite diameter, Chebyshev constant and Wiener energy in the abstract linear potential analytic setting pioneered by Choquet, Fuglede and Ohtsuka. It turns out that, whenever the potential theoretic kernel has the maximum principle, then all these quantities are equal for all compact sets. For continuous kernels even the converse statement is true: if the Chebyshev constant of any compact set coincides with its transfinite diameter, the kernel must satisfy the maximum principle. An abundance of examples is provided to show the sharpness of the results.