Universal optimal configurations for the $p$frame potentials
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 $N$ unitnorm 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.
 Publication:

arXiv eprints
 Pub Date:
 February 2019
 arXiv:
 arXiv:1902.03505
 Bibcode:
 2019arXiv190203505C
 Keywords:

 Computer Science  Information Theory;
 Mathematics  Functional Analysis