# Codes and Expansions (CodEx) Seminar

## Kasso A. Okoudjou (Tufts University)Minimizing the $p$ frame potentials

Given $d \geq 2$, $p \in (0,\infty]$, and $N \geq 2$, the discrete $p$ frame potential is defined by
$FP_{p,d,N}(\Phi) := \sum_{k,\ell=1}^N | \langle \varphi_k, \varphi_\ell \rangle |^p$
where $\Phi = \{ \varphi_k \}_{k=1}^N \subset S^{d-1}$, and $S^{d-1}$ is the unit sphere in $\mathbb{R}^d$. This function can be viewed as the restriction to discrete probability measures, of the continuous $p$-frame potentials defined by
$\iint_{S^{d-1} \times S^{d-1}} | \langle x, y \rangle |^p d\mu(x) d\mu(y)$
where $\mu$ runs over the set of all Borel probability measures defined on $S^{d-1}$. In this talk, we will focus on the relation between the minimizers of these two functionals. In particular, we will report on ongoing work in dimension $d = 2$ where the classification of the minimizers of these functionals is still incomplete.