Universal optimal configurations for the $p$-frame potentials

TL;DR

This paper studies the minimizers of p-frame potentials for collections of unit vectors, revealing universal optimal configurations for large p in two dimensions and providing conjectures for higher dimensions based on numerical experiments.

Contribution

It establishes the unique minimizer for large p in 2D and shows its universality, extending understanding of p-frame potentials beyond known cases.

Findings

Identifies the unique minimizer for large p in 2D.

Shows the minimizer is universal across a range of energy functions.

Provides numerical conjectures for higher dimensions and p in (0,2).

Abstract

Given $d, N\geq 2$ and $p\in (0, \infty]$ we consider a family of functionals, the $p$-frame potentials FP$_{p, N, d}$, defined on the set of all collections of $N$ unit-norm vectors in $\mathbb R^d$. For the special case $p=2$ and $p=\infty$, both the minima and the minimizers of these potentials have been thoroughly investigated. In this paper, we investigate the minimizers of the functionals FP$_{p, N, d}$, by first establishing some general properties of their minima. Thereafter, we focus on the special case $d=2$, for which, surprisingly, not much is known. One of our main results establishes the unique minimizer for big enough $p$. Moreover, this minimizer is universal in the sense that it minimizes a large range of energy functions that includes the $p$-frame potential. We conclude the paper by reporting some numerical experiments for the case $d\geq 3$, $N=d+1$, $p\in (0, 2)$. These experiments lead to some conjectures that we pose.