More on the differentiability of convex functions
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- by Maria Elena Verona PDF
- Proc. Amer. Math. Soc. 103 (1988), 137-140 Request permission
Abstract:
Let $C$ be a closed, convex set in a topological vector space $X$ such that $NS(C)$, the set of its nonsupport points, is nonempty (this is always the case if $X$ is Banach separable; if $X$ is Fréchet, $NS\left ( C \right )$ is residual in $C$). If $X$ is normed, we prove that any locally Lipschitz, convex real function $f$ on $C$ is subdifferentiable on $NS\left ( C \right )$. If in addition $X$ is Banach separable, we prove that $f$ is smooth on a residual subset of $NS\left ( C \right )$.References
- A. Brøndsted and R. T. Rockafellar, On the subdifferentiability of convex functions, Proc. Amer. Math. Soc. 16 (1965), 605–611. MR 178103, DOI 10.1090/S0002-9939-1965-0178103-8
- E. E. Floyd and V. L. Klee, A characterization of reflexivity by the lattice of closed subspaces, Proc. Amer. Math. Soc. 5 (1954), 655–661. MR 63020, DOI 10.1090/S0002-9939-1954-0063020-7
- Richard B. Holmes, Geometric functional analysis and its applications, Graduate Texts in Mathematics, No. 24, Springer-Verlag, New York-Heidelberg, 1975. MR 0410335
- V. L. Klee Jr., Convex sets in linear spaces, Duke Math. J. 18 (1951), 443–466. MR 44014
- R. R. Phelps, Some topological properties of support points of convex sets, Israel J. Math. 13 (1972), 327–336 (1973). MR 328558, DOI 10.1007/BF02762808
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 103 (1988), 137-140
- MSC: Primary 58C20; Secondary 26B25, 49A51, 90C25
- DOI: https://doi.org/10.1090/S0002-9939-1988-0938657-3
- MathSciNet review: 938657