## School of Mathematical and Statistical Sciences Faculty Publications and Presentations

#### Title

Repeated Minimizers of $p$-Frame Energies

Article

11-18-2020

#### Abstract

For a collection of $N$ unit vectors ${X}=\{x_i\}_{i=1}^N$, define the $p$-frame energy of ${X}$ as the quantity $\sum_{i\neq j} |\langle x_i,x_j \rangle|^p$. In this paper, we connect the problem of minimizing this value to another optimization problem, thus giving new lower bounds for such energies. In particular, for $p<2$, we prove that this energy is at least $2(N-d) p^{-\frac p 2} (2-p)^{\frac {p-2} 2}$ which is sharp for $d\leq N\leq 2d$ and $p=1$. We also prove that for \$1\leq m

#### Publication Title

SIAM Journal on Discrete Mathematics

#### DOI

10.1137/19M1282702

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