Repeated minimizers of $p$-frame energies

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 ${\sum }_{i\ne j}|⟨x_i,x_j⟩{|}^p$. 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 $d\le N\le 2d$ and $p=1$. We prove that for $1\le m<d$, 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.

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