We derive $C^2-$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 well-known 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 $C^2-$condition for strict convexity to a characterization. Several examples illustrate the results.
Schaedae Informaticae, Vol. 21 (2012), 55-63.