On faces of convex sets of positive semidefinite matrices

Question

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 $C_p$?

Answer 1

The space $H_+$ of positive semidefinite Hermitian $n\times n$ matrices is invariant under the action by conjugation of the group $SU(n)$, which is isometric for the $\ell^2$ metric. Here the space of diagonal matrices with non-negative entries is a submanifold which meets each orbit orthogonally: this is called a polar action. The orbit space is the Weyl chamber of diagonal matrices with (say) decreasingly ordered non-negative entries. The faces of the cone are not the orbits, but the strata of the orbit type stratification. Each connected component of such an orbit type stratum is projected onto a face of the Weyl chamber, so its dimension is the dimension of the face of the Weyl chamber plus the dimension of a typical orbit (= $SU(n)/\text{Isotropy group}$) which is easy to compute.

See 7.1, section 30, and 34.16 of here for background and more information