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arXiv · 2411.00290

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

S. Borodachov·P. Boyvalenkov·P. Dragnev. D. Hardin. E. Saff·M. Stoyanova
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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.

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S. Borodachov, P. Boyvalenkov, P. Dragnev. D. Hardin. E. Saff, M. Stoyanova. 2024-11-01. Bounds on Discrete Potentials of Spherical (k,k)-Designs. https://arxiv.org/abs/2411.00290

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