Optimal Discrete Measures for Potentials: Polarization (Chebyshev) Constants

Abstract

This chapter investigates optimal discrete measures from the perspective of a max-min problem for potentials on a given compact set A. More precisely, for a kernel $K:A\times A \rightarrow \mathbb {R}\cup \{+\infty \}$, the so-called polarization (or Chebyshev) problem is the following: determine N-point configurations $\{x_j\}_{j=1}^N$ on A so that the minimum of $\sum _{j=1}^NK(x, x_j)$ for $x\in A$ is as large as possible. Such optimization problems relate to the following practical question: if $K(x, x_j)$ denotes the amount of a substance received at x due to an injector of the substance located at $x_j$, what is the smallest number of like injectors and their optimal locations on A so that a prescribed minimal amount of the substance reaches every point of A?