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

## TL;DR

This paper introduces and analyzes the unconstrained polarization problem in Euclidean space, revealing how optimal configurations relate to the set $A$ and deriving asymptotics for Riesz kernels, with special cases on spheres and balls.

## Contribution

It establishes fundamental properties of the unconstrained polarization problem, compares it to the constrained case, and derives asymptotic results for Riesz kernels on various sets.

## Key findings

- Optimal configurations concentrate near set $A$ for certain Riesz kernels.
- Asymptotic polarization values match those of constrained problems on rectifiable sets.
- Special cases analyzed include configurations on spheres and balls.

## 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.