A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions

Authors:
J. M. Borwein and D. Preiss

Journal:
Trans. Amer. Math. Soc. **303** (1987), 517-527

MSC:
Primary 49A27; Secondary 46B20, 46G05, 49A51, 49A52, 58C20, 90C25, 90C48

DOI:
https://doi.org/10.1090/S0002-9947-1987-0902782-7

MathSciNet review:
902782

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Abstract | References | Similar Articles | Additional Information

Abstract: We show that, typically, lower semicontinuous functions on a Banach space densely inherit lower subderivatives of the same degree of smoothness as the norm. In particular every continuous convex function on a space with a Gâteaux (weak Hadamard, Fréchet) smooth renorm is densely Gâteaux (weak Hadamard, Fréchet) differentiable. Our technique relies on a more powerful analogue of Ekeland's variational principle in which the function is perturbed by a quadratic-like function. This "smooth" variational principle has very broad applicability in problems of nonsmooth analysis.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1987-0902782-7

Keywords:
Weak Asplund spaces,
subderivatives,
renorms,
nonsmooth analysis,
Ekeland's principle,
proximal normals

Article copyright:
© Copyright 1987
American Mathematical Society