Optimizers of three-point energies and nearly orthogonal sets

Bilyk, Dmitriy; Ferizović, Damir; Glazyrin, Alexey; Matzke, Ryan; Park, Josiah; Vlasiuk, Oleksandr

doi:10.1090/proc/16868

Optimizers of three-point energies and nearly orthogonal sets
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by Dmitriy Bilyk, Damir Ferizović, Alexey Glazyrin, Ryan W. Matzke, Josiah Park and Oleksandr Vlasiuk;

Proc. Amer. Math. Soc. 152 (2024), 4015-4033

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

Published electronically: July 31, 2024

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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 Erdős. As the outcome, we provide a new solution of the Erdős 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 $\mathbb {S}^{d-1}$.

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Optimizers of three-point energies and nearly orthogonal sets
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Abstract: