Eigenvalue-constrained faces

We characterize the exposed faces of convex sets of symmetric matrices, invariant under orthogonal similarity (UTCU = C for all orthogonal U). Such sets are exactly those determined by eigenvalue constraints: typical examples are the positive semidefinite cone and the unit balls of common matrix norms. The set of all diagonal matrices in is known to be convex if and only if is, and is invariant under the group of permutations (acting on diagonal entries). We show how any exposed face of is naturally associated with an exposed face of , by relating the stabilizer groups of the two faces.