On the Dimension of a Face Exposed by Proper Separation of Convex Polyhedra

Whenever two nonempty convex polyhedra can be properly separated, a separating hyperplane may be chosen to contain a face of either polyhedron. It is demonstrated that, in fact, one or the other of the polyhedra admits such an exposed face having dimension no smaller than approximately half the larger dimension of the two polyhedra. An example shows that the bound on face dimension is optimal, and a linear programming representation of the problem is given.