  The state space of an operator system of $n$-by-$n$ matrices has, in a sense,
many normal cones. Merely this convex geometrical property implies smoothness
qualities and a clustering property of exposed faces. The latter holds since
each exposed face is an intersection of maximal exposed faces. An isomorphism
translates these results to the lattice of ground state projections of the
operator system. We work on minimizing the assumptions under which a convex set
has the mentioned properties.
