Min-Max Polarization for Certain Classes of Sharp Configurations on the Sphere

2022/03/25 by Sergiy Borodachov, Borodachov, Sergiy
Materials Science · Mathematics · #05B30 #31B15 #33D45 #42C05 #51E30 #52C35 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Approximation and Integration #Quasicrystal Structures and Properties

paper · pdf · doi:10.48550/arxiv.2203.13756

openalex publication_date 2022/03/25 · openalex created_date 2022/04/03 · openalex updated_date 2026/07/28

Abstract

We consider the problem of finding an N-point configuration on the sphere S^d⊂ \RR^d+1 with the smallest absolute maximum value over S^d of its total potential. The potential induced by each point \bf y in a given configuration at a point \bf x∈ S^d is f\(|\bf x-\bf y|²\), where f is continuous on [0,4] and completely monotone on (0,4], and |\bf x-\bf y| is the Euclidean distance between points~\bf x and \bf y. We show that any sharp point configuration \OLω_N on S^d, which is antipodal or is a spherical design of an even strength is a solution to this problem. We also prove that the absolute maximum over S^d of the potential of any such configuration \OLω_N is attained at points of \OLω_N.

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