Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

Luděk Zajíček

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

zajicek@karlin.mff.cuni.cz

Keywords

  • Semiconvex function
  • strongly paraconvex function
  • generalized subdifferentials
  • suprema of smooth functions

Mathematics Subject Classification

26B25; 46T99

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Suggested citation

J. Duda, L. Zajíček. “Semiconvex Functions: Representations as Suprema of Smooth Functions and Extensions.” Journal of Convex Analysis 16 (2009), No. 1, 239–260. https://doi.org/10.68381/jca16013

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Published by Heldermann Verlag, 2009. Rights now held by Banach Press LLC.