Unconstrained Polarization (Chebyshev) Problems: Basic Properties and Riesz Kernel Asymptotics

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.