Twice Differentiable Characterizations of Convexity Notions for Functions on Full Dimensional Convex Sets

Oliver Stein

Abstrakt

IWe derive C2 −characterizations for convex, strictly convex, as well as strongly convex functions on full dimensional convex sets. In the cases of convex and strongly 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.

Słowa kluczowe: convexity, strict convexity, strong convexity, true convexity, differentiable characterization, full dimensional convex set.
References

Hiriart-Urruty J.-B., Lemar´echal C.; Fundamentals of Convex Analysis, Springer 2001.

Jongen H.Th., Schmidt P.G.; Analysis, Lecture Notes, Aachener Beitr¨age zur Mathematik 19, Wissenschaftsverlag, Mainz 1998.

Rockafellar R.T.; Convex Analysis, Princeton University Press 1970.

Rockafellar R.T., Wets R.; Variational Analysis, Springer 1998.

Shikhman V., Stein O.; On jet convex functions and their tensor products, Optimization 61, 2012, pp. 717–731.

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