School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Optimizers of three-point energies and nearly orthogonal sets

Document Type

Article

Publication Date

7-2024

Abstract

This paper is devoted to spherical measures and point configurations optimizing three-point energies. Our main goal is to extend the classic optimization problems based on pairs of distances between points to the context of three-point potentials. In particular, we study three-point analogues of the sphere packing problem and the optimization problem for p-frame energies based on three points. It turns out that both problems are inherently connected to the problem of nearly orthogonal sets by Erd ̋os. As the outcome, we provide a new solution of the Erd ̋os problem from the three-point packing perspective. We also show that the orthogonal basis uniquely minimizes the p-frame three-point energy when 0 < p < 1 in all dimensions. The arguments make use of multivariate polynomials employed in semidefinite programming and based on the classical Gegenbauer polynomials. For p = 1, we completely solve the analogous problem on the circle. As for higher dimensions, we show that the Hausdorff dimension of minimizers is not greater than d − 2 for measures on Sd−1.

Comments

Original published version available at https://doi.org/10.1090/proc/16868

Publication Title

Proceedings of the American Mathematical Society

DOI

https://doi.org/10.1090/proc/16868

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