Twice differentiable characterizations of convexity notions for functions on full dimensional convex sets

We derive C2−characterizations for convex, strictly convex, as well as uniformly convex functions on full dimensional convex sets. In the cases of convex and uniformly convex functions this weakens the wellknown openness assumption on the convex sets. We also show that, in a certain sense, the full dimensionality assumption cannot be weakened further. In the case of strictly convex functions we weaken the well-known sufficient C2−condition for strict convexity to a characterization. Several examples illustrate the results.