On the Regularity of the Convexification Operator on a Compact Set

Let $co_X(\cdot)$ denote the convexification operator on bounded real functions on a convex compact set X. Several necessary and sufficient conditions for the operator $co_X(\cdot)$ to preserve continuity and uniformly Lipschitz continuity are established. In the special case of a finite dimensional topological vector space, it is shown that (1) the preservation of continuity is equivalent to the closeness of the set of faces of X and (2) the uniform preservation of Lipschitz continuity is equivalent to X being a polytope