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

Borwein, J. M.; Preiss, D.

doi:10.1090/S0002-9947-1987-0902782-7

A smooth variational principle with applications to subdifferentiability and to differentiability of convex functions
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by J. M. Borwein and D. Preiss

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

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

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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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