Eigenvalue-constrained Faces

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