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Journal of Convex Analysis 16 (2009), No. 1, 239--260
Copyright Heldermann Verlag 2009

Semiconvex Functions: Representations as Suprema of Smooth Functions and Extensions

Jakub Duda
PIRA Energy Group, 3 Park Ave Fl 26, New York, NY 10016, U.S.A.

Ludek Zajícek
Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic

We prove results on representations of semiconvex functions with an arbitrary modulus (equivalently: strongly paraconvex functions) in superreflexive Banach spaces as suprema of families of differentiable functions. Also, results on extensions of semiconvex functions are proved. Further, characterizations of semiconvex functions by uniform Fréchet subdifferentiability and (global) [α]-subdifferentiability are given. We also show that weakly convex functions in Nurminskii's sense coincide with locally semiconvex functions.

Keywords: Semiconvex function, strongly paraconvex function, generalized subdifferentials, suprema of smooth functions.

MSC: 26B25; 46T99

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