## 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 PDF
- Trans. Amer. Math. Soc.
**303**(1987), 517-527 Request permission

## 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.## References

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*Proximal analysis and boundaries of closed sets in Banach space, Part*2:

*Applications*, Canad. Math. J (to appear). —,

*Subderivatives and nonsmooth analysis*(to appear).

*Frèchet derivatives of Lipschitz functions*(to appear).

## Additional Information

- © Copyright 1987 American Mathematical Society
- 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