Mean value property and subdifferential criteria for lower semicontinuous functions

Aussel, Didier; Corvellec, Jean-Noël; Lassonde, Marc

doi:10.1090/S0002-9947-1995-1307998-0

Mean value property and subdifferential criteria for lower semicontinuous functions
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by Didier Aussel, Jean-Noël Corvellec and Marc Lassonde

Trans. Amer. Math. Soc. 347 (1995), 4147-4161

DOI: https://doi.org/10.1090/S0002-9947-1995-1307998-0

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Abstract:

We define an abstract notion of subdifferential operator and an associated notion of smoothness of a norm covering all the standard situations. In particular, a norm is smooth for the Gâteaux (Fréchet, Hadamard, Lipschitz-smooth) subdifferential if it is Gâteaux (Fréchet, Hadamard, Lipschitz) smooth in the classical sense, while on the other hand any norm is smooth for the Clarke-Rockafellar subdifferential. We then show that lower semicontinuous functions on a Banach space satisfy an Approximate Mean Value Inequality with respect to any subdifferential for which the norm is smooth, thus providing a new insight on the connection between the smoothness of norms and the subdifferentiability properties of functions. The proof relies on an adaptation of the "smooth" variational principle of Borwein-Preiss. Along the same vein, we derive subdifferential criteria for coercivity, Lipschitz behavior, conemonotonicity, quasiconvexity, and convexity of lower semicontinuous functions which clarify, unify and extend many existing results for specific subdifferentials.

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