# Repeated minimizers of $p$-frame energies

## TL;DR

This paper investigates the minimization of $p$-frame energies for collections of unit vectors, establishing new bounds, identifying optimal configurations for specific parameters, and proposing conjectures for broader cases.

## Contribution

It introduces new lower bounds for $p$-frame energies, characterizes minimizers for certain dimensions and parameters, and connects the problem to alternative optimization formulations.

## Key findings

- Established sharp lower bounds for $p$-frame energies when $p<2$ and $N \,\leq \, 2d$.
- Identified repeated orthonormal basis constructions as minimizers for specific $p$ intervals.
- Proposed conjectures on the minimization problem for general energies and configurations.

## Abstract

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.