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

Oliver Stein

Abstract. 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 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 C²-condition for strict convexity to a characterization. Several examples illustrate the results.

 

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