Optimal measures for multivariate geometric potentials

Authors

Dmitriy Bilyk
Damir Ferizovic
Alexey Glazyrin, The University of Texas Rio Grande ValleyFollow
Ryan Matzke
Josiah Park
Oleksandr Vlasiuk

Document Type

Article

Publication Date

2025

Abstract

We study measures and point configurations optimizing energies based on multivariate potentials. The emphasis is put on potentials defined by geometric characteristics of sets of points, which serve as multi-input generalizations of the well-known Riesz potentials for pairwise interaction. One of such potentials is volume squared of the simplex with vertices at the k ≥ 3 given points: we show that the arising energy is maximized by balanced isotropic measures, in contrast to the classical two-input energy. These results are used to obtain interesting geometric optimality properties of the regular simplex. As the main machinery, we adapt the semidefinite programming method to this context and establish relevant versions of the kpoint bounds.

Comments

Copyright Indiana University Mathematics Journal

Recommended Citation

Bilyk, Dmitriy, Damir Ferizovic, Alexey Glazyrin, Ryan Matzke, Josiah Park, and Oleksandr Vlasiuk. 2025. “Optimal Measures for Multivariate Geometric Potentials.” Indiana University Mathematics Journal 74 (3): 721–57.https://doi.org/10.1512/iumj.2025.74.60289

First Page

721

Last Page

757

Publication Title

Indiana University Mathematics Journal