Bounds on Discrete Potentials of Spherical (k,k)-Designs

2024/11/01 by Borodachov, S., Boyvalenkov, P., Saff, P. Dragnev. D. Hardin. E. +1 · 1 citation
#05B30 #52C35 #Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2411.00290

Abstract

We derive universal lower and upper bounds for max-min and min-max problems (also known as polarization) for the potential of spherical (k,k)-designs and provide certain examples, including unit-norm tight frames, that attain these bounds. The universality is understood in the sense that the bounds hold for all spherical (k,k)-designs and for a large class of potential functions, and the bounds involve certain nodes and weights that are independent of the potential. When the potential function is h(t)=t^2k, we prove an optimality property of the spherical (k,k)-designs in the class of all spherical codes of the same cardinality both for max-min and min-max potential problems.

Cited by

• Real and complex spherical designs and their Gramian

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