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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

More on the differentiability of convex functions


Author: Maria Elena Verona
Journal: Proc. Amer. Math. Soc. 103 (1988), 137-140
MSC: Primary 58C20; Secondary 26B25, 49A51, 90C25
MathSciNet review: 938657
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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)$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1988-0938657-3
PII: S 0002-9939(1988)0938657-3
Article copyright: © Copyright 1988 American Mathematical Society