The minimizers of the $p$-frame potential

For any positive real number $p$, the $p$-frame potential of $N$ unit vectors $X:=\{\mathbf x_1,\ldots,\mathbf x_N\}\subset \mathbb R^d$ is defined as ${\rm FP}_{p,N,d}(X)=\sum_{i\neq j}|\langle \mathbf x_i,\mathbf x_j\rangle |^p$. In this paper, we focus on the special case $N=d+1$ and establish the unique minimizer of ${\rm FP}_{p,d+1,d}$ 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.