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

2019/02/22 by Hardin, Douglas P., Petrache, Mircea, Saff, Edward B. · 1 citation
#30C80 (Secondary) #31C15 #31C20 (Primary) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1902.08497

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.

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• The polarization problem for the honeycomb structure

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