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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
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.

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Article copyright: © Copyright 1994 American Mathematical Society