Polarization constants

In potential theory and optimization, polarization constants (also known as Chebyshev constants) are solutions to a max-min problem for potentials . More precisely, for a compact set $A$ and kernel $K:A\times A\to \mathbb {R} \cup \{+\infty \}$, the discrete polarization problem is the following: determine $N$-point configurations $\{x_{j}\}_{j=1}^{N}$ on $A$ so that the minimum of $\sum _{j=1}^{N}K(x,x_{j})$ 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 $A$ ?

The Chebyshev nomenclature for this max-min problem emanates from the case when $K$ is the logarithmic kernel, $K(x,y)=\log |x-y|^{-1},$ for when $A$ is a subset of the complex plane, the problem is equivalent to finding the constrained $N$-th degree Chebyshev polynomial for $A$; that is, the monic polynomial in the complex variable $z$ with all its zeros on $A$ having minimal uniform norm on $A$.

If $A$ is the unit circle in the plane and $K(x,y)=|x-y|^{-s}$, $s>0$, then $N$ equally spaced points on the circle solve the $N$ point polarization problem.

For connections with minimum energy problems, particularly the Thomson problem, see and