Repeated minimizers of $p$-frame energies

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 $sum_{i eq j} |langle x_i,x_j rangle|^p$. 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 $dleq Nleq 2d$ and $p=1$. We prove that for $1leq m<d$, a repeated orthonormal basis construction of $N=d+m$ vectors minimizes the energy over an interval, $pin[1,p_m]$, and demonstrate an analogous result for all $N$ in the case $d=2$. Finally, in connection, we give conjectures on these and other energies.