  Given $d, N\geq 2$ and $p\in (0, \infty]$ we consider a family of
functionals, the $p$-frame potentials FP$_{p, N, d}$, defined on the set of all
collections of $N$ unit-norm vectors in $\mathbb R^d$. For the special case
$p=2$ and $p=\infty$, both the minima and the minimizers of these potentials
have been thoroughly investigated. In this paper, we investigate the minimizers
of the functionals FP$_{p, N, d}$, by first establishing some general
properties of their minima. Thereafter, we focus on the special case $d=2$, for
which, surprisingly, not much is known. One of our main results establishes the
unique minimizer for big enough $p$. Moreover, this minimizer is universal in
the sense that it minimizes a large range of energy functions that includes the
$p$-frame potential. We conclude the paper by reporting some numerical
experiments for the case $d\geq 3$, $N=d+1$, $p\in (0, 2)$. These experiments
lead to some conjectures that we pose.
