Prox-Regularity of Functions and Sets in Banach Spaces

In this paper, we define the prox-regularity for functions on Banach spaces by adapting the original definition in R ⁿ . In this context, we establish a subdifferential characterization and show that qualified convexly C ^1,+-composite functions and primal lower nice functions belong to this class, as already known in the setting of Hilbert spaces. We also study, in a geometrical point of view, the epigraphs of prox-regular functions. The subdifferential characterization allows us to show that some Moreau-envelope-like regularizations of such functions are of class C ¹ in the context of certain uniformly convex spaces.