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 positive semidefinite cone $H_+ = \text{conv} \{ xx^* \}$ in the real vector space of Hermitian matrices 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, is there a way to characterize the smallest face containing a given point, and find its dimension? Similarly as for $H_+$, every matrix $A \in C_p$ certainly belongs to the face given by

$$ \text{conv} \{ vv^* : \|v\|_p \leq 1,\, \text{ker}(A) \subset \text{ker}(vv^*) \} $$

but now this face is not necessarily the smallest (take e.g. the extremal points of $C_p$, which we know to lie in a 0-dimensional face). Could it be that this face *is* still the smallest for any non-extremal point?

I am curious about this because an interesting corollary of the characterization of the faces of $H_+$ is that we have $\text{dim}\, F(H_+, A) = \text{rank}(A)^2$, which means that there 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 $C_p$?

90(1987) 81–88, esp. cor. 3.7. $\endgroup$ – Gro-Tsen Feb 17 '18 at 4:43