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A note on the differentiability of convex functions


Authors: Cong Xin Wu and Li Xin Cheng
Journal: Proc. Amer. Math. Soc. 121 (1994), 1057-1062
MSC: Primary 46G05; Secondary 49J50
DOI: https://doi.org/10.1090/S0002-9939-1994-1207535-X
MathSciNet review: 1207535
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Abstract: Every real-valued convex and locally Lipschitzian function f defined on a nonempty closed convex set D of a Banach space E is the local restriction of a convex Lipschitzian function defined on E. Moreover, if E is separable and $ \operatorname{int} D \ne \emptyset $, then, for each Gateaux differentiability point x $ ( \in \operatorname{int} D)$ of f, there is a closed convex set $ C \subset \operatorname{int} D$ with the nonsupport points set $ N(C) \ne \emptyset $ and with $ x \in N(C)$ such that $ {f_C}$ (the restriction of f on C) is Fréchet differentiable at x.


References [Enhancements On Off] (What's this?)

  • [1] J. Rainwater, Yet more on the differentiability of convex functions, Proc. Amer. Math. Soc. 103 (1988), 773-777. MR 947656 (89m:46081)
  • [2] M. E. Verona, More on the differentiability of convex functions, Proc. Amer. Math. Soc. 103 (1988), 137-140. MR 938657 (89f:58016)
  • [3] J. B. Hiriart Urruty, Lipschitz r-continuity of the approximate subdifferential of a convex function, Math. Scand. 47 (1980), 123-134. MR 600082 (82c:58007)
  • [4] P. J. Laurent, Approximation et optimisation, Hermann, Paris, 1972. MR 0467080 (57:6947)
  • [5] E. Asplund, Fréchet differentiability of convex functions, Acta. Math. 121 (1968), 31-47. MR 0231199 (37:6754)
  • [6] S. Fitzpatrick and R. R. Phelps, Bounded approximants to monotone operators on Banach spaces (to appear). MR 1191009 (93j:47076)
  • [7] D. Preiss, R. R. Phelps, and I. Namioka, Smooth Banach spaces, weak Asplund spaces and monotone or USCO mappings (to appear). MR 1120220 (92h:46021)
  • [8] J. Lindenstrauss and L. Tzafiri, Classical Banach spaces. I, Springer-Verlag, New York, 1977. MR 0500056 (58:17766)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1994-1207535-X
Article copyright: © Copyright 1994 American Mathematical Society

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