Repeated Minimizers of $p$-Frame Energies

Authors

Alexey Glazyrin, The University of Texas Rio Grande Valley
Josiah Park

Document Type

Article

Publication Date

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

Comments

© 2020, Society for Industrial and Applied Mathematics Read More: https://epubs.siam.org/doi/10.1137/19M1282702

Recommended Citation

Glazyrin, A., & Park, J. (2020). Repeated Minimizers of $p$-Frame Energies. SIAM Journal on Discrete Mathematics, 34(4), 2411–2423. https://doi.org/10.1137/19M1282702

Publication Title

SIAM Journal on Discrete Mathematics

DOI

10.1137/19M1282702