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.