Unconstrained polarization (Chebyshev) problems: basic properties and Riesz kernel asymptotics

Abstract

We introduce and study the unconstrained polarization (or Chebyshev) problem which requires to find an $N$-point configuration that maximizes the minimum value of its potential over a set $A$ in $p$-dimensional Euclidean space. This problem is compared to the constrained problem in which the points are required to belong to the set $A$. We find that for Riesz kernels $1/|x-y|^s$ with $s>p-2$ the optimum unconstrained configurations concentrate close to the set $A$ and based on this fundamental fact we recover the same asymptotic value of the polarization as for the more classical constrained problem on a class of $d$-rectifiable sets. We also investigate the new unconstrained problem in special cases such as for spheres and balls. In the last section we formulate some natural open problems and conjectures.