Log-optimal (d + 2)-configurations in d–dimensions

Authors

Peter D. Dragnev
Oleg R. Musin, The University of Texas Rio Grande Valley

Document Type

Article

Publication Date

2-3-2023

Abstract

We enumerate and classify all stationary logarithmic configurations of d + 2 points on the unit sphere in d–dimensions. In particular, we show that the logarithmic energy attains its local minima at configurations that consist of two orthogonal to each other regular simplexes of cardinality m and n. The global minimum occurs when m = n if d is even and m = n + 1 otherwise. This characterizes a new class of configurations that minimize the logarithmic energy on Sd−1 for all d. The other two classes known in the literature, the regular simplex (d + 1 points on Sd−1) and the cross-polytope (2d points on Sd−1), are both universally optimal configurations.

Comments

Copyright 2023 by the author(s) under Creative Commons Attribution-NonCommercial 3.0 License (CC BY NC 3.0)

Recommended Citation

Dragnev, Peter, and Oleg Musin. "Log-optimal (𝑑+ 2)-configurations in 𝑑–dimensions." Transactions of the American Mathematical Society, Series B 10.05 (2023): 155-170. https://doi.org/10.1090/btran/118

Creative Commons License

This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 License

First Page

155

Last Page

170

Publication Title

Transactions of the American Mathematical Society, Series B

DOI

10.1090/btran/118