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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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
MathSciNet review: 902782
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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