The minimizers of the $p$-frame potential

Abstract

For any positive real number $p$, the $p$-frame potential of $N$ unit vectors $X:=\{{\mathbf{x}}_1,\dots ,{\mathbf{x}}_N\}\subset {\mathbb{R}}^d$ is defined as ${\mathrm{FP}}_{p,N,d}(X)={\sum }_{i\ne j}|⟨{\mathbf{x}}_i,{\mathbf{x}}_j⟩{|}^p$. In this paper, we focus on the special case $N=d+1$ and establish the unique minimizer of ${\mathrm{FP}}_{p,d+1,d}$ for $p\in (0,2)$. Our results completely solve the minimization problem of $p$-frame potential when $N=d+1$, which confirms a conjecture posed by Chen, Gonzales, Goodman, Kang and Okoudjou.

18 pages