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

Oliver Stein (stein***at***kit.edu)

Abstract: 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.

Keywords: Convexity, strict convexity, true convexity, differentiable characterization, full dimensional convex set

Category 1: Convex and Nonsmooth Optimization (Convex Optimization )

Citation: Schaedae Informaticae, Vol. 21 (2012), 55-63.

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Entry Submitted: 08/01/2011
Entry Accepted: 08/01/2011
Entry Last Modified: 03/27/2013

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