Abstract

We prove that a locally Lipschitz function on an open subset GG of an Asplund space XX, whose restrictions to ``many lines'' are essentially smooth (i.e., almost everywhere strictly differentiable), is generically Fréchet differentiable on XX. In this way we obtain new proofs of known Fréchet differentiability properties of approximately convex functions, Lipschitz regular functions, saddle (or biconvex) Lipschitz functions, and essentially smooth functions (in the sense of Borwein and Moors), and also some new differentiability results (e.g., for partially DC functions). We show that classes of functions Seg(G)\Se_e^{g}(G) and Reg(G)\Rc_e^{g}(G) (defined via linear essential smoothness) are respectively larger than classes Se(G)\Se_e(G) (of essentially smooth functions) and Re(G)\Rc_e(G) studied by Borwein and Moors, and have also nice properties. In particular, we prove that members of Seg(G)\Se_e^{g}(G) are uniquely determined by their Clarke subdifferentials. We also show the inclusion Se(G)⊂Re(G)\Se_e(G) \subset \Rc_e(G) for Borwein-Moors classes.

Author information

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

Luděk Zajíček

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

zajicek@karlin.mff.cuni.cz

Keywords

  • Generic Fréchet differentiability
  • essentially smooth functions
  • separable reduction

Mathematics Subject Classification

46G05; 46T20

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Suggested citation

L. Zajíček. “Generic Fréchet Differentiability on Asplund Spaces via A.E. Strict Differentiability on Many Lines.” Journal of Convex Analysis 19 (2012), No. 1, 23–48. https://doi.org/10.68381/jca19002

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Published by Heldermann Verlag, 2012. Rights now held by Banach Press.