A face $ F$ of a convex set $ C$ is a set such that, if $ x \in F$ is a convex combination of other elements in $ C$ , then they must also be in $ F$ . I will denote by $ F(C,x)$ the smallest face of $ C$ which contains $ x$ .

The faces of the complex positive semidefinite cone $ H_+$ are well characterized, and we know that $ F(H_+, A) = \left\{ B : \text{ker}(A) \subset \text{ker}(B) \right\}$ .

I am interested in the faces of subsets of this cone of the form

$ $ C_p = \text{conv} \{ vv^* : \|v\|_p \leq 1 \} $ $

where conv denotes the convex hull and $ \|\|_p$ is any $ \ell_p$ norm. I am specifically looking at $ p=1$ and $ p=\infty$ , but more general results could also be interesting.

In particular, are there some ways to get bounds on the dimensions of the corresponding faces? An interesting corollary of the characterization of the faces of $ H_+$ is that we have $ \text{dim}\, F(H_+, A) = rank(A)^2$ , which means that are faces of dimension 1 (the extreme rays generated by rank-one matrices) and faces of dimension 4 (generated by rank-two matrices), but no faces of dimension 2 or 3. Can anything similar be said about the sets $ C_p$ ?