Frame Potentials and Orthogonal Vectors

An extension is given of a recent result of Glazyrin, showing that an orthonormal basis $\{e_i{\}}_{i=1}^d$ joined with the vectors $\{e_j{\}}_{j=1}^m$, where $1\le m<d$ minimizes the $p$-frame potential for $p\in [1,2\log \frac{2m+1}{2m}\mathrm{/}\log \frac{m+1}{m}]$ over all collections of $N=d+m$ vectors $\{x_1,\dots ,x_N\}$ in ${\mathbb{S}}^{d-1}$.