Repeated minimizers of $p$-frame energies

• Main Authors: Glazyrin, Alexey, Park, Josiah
• Format: Journal Article
• Language: English
• Published: 18.01.2019
• Subjects: Mathematics - Functional Analysis; Mathematics - Metric Geometry
• DOI: 10.48550/arxiv.1901.06096

SIAM Journal on Discrete Mathematics, 34, no. 4 (2020), 2411-2423 For a collection of $N$ unit vectors $\mathbf{X}=\{x_i\}_{i=1}^N$, define the $p$-frame energy of $\mathbf{X}$ 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, so 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 prove that for $1\leq m<d$, a repeated orthonormal basis construction of $N=d+m$ vectors minimizes the energy over an interval, $p\in[1,p_m]$, and demonstrate an analogous result for all $N$ in the case $d=2$. Finally, in connection, we give conjectures on these and other energies.