  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 $\sum_{i\neq j} |\langle
x_i,x_j \rangle|^p$. 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 $d\leq N\leq
2d$ and $p=1$. We prove that for $1\leq m<d$, 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.
