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Operator systems and convex sets with many normal cones

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.