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Journal of Convex Analysis 29 (2022), No. 4, 1225--1250
Copyright Heldermann Verlag 2022

Subconvex Functions, Relations and Sets

Jean-Paul Penot
Sorbonne Universities, UPMC Univ Paris 6, CNRS -- Lab. J.-L. Lions, Paris, France

We consider a class of subsets of a Banach space whose normal cones satisfy a kind of uniformity property. Here the normal cone is associated with a general subdifferential as defined in an appendix for the sake of clarity. We examine to what extent this property depends on the choice of the subdifferential. We relate this property to approximate convexity in the sense of Ngai, Luc and Théra, subsmoothness in the sense of Aussel, Daniilidis, Thibault and super-regularity in the sense of Lewis, Luke and Malick, showing coincidence in appropriate spaces. This class of sets is larger than the classes of amenable subsets and prox-regular subsets. We also explore related properties for functions and multifunctions. We devise some permanence properties under usual operations.

Keywords: Approximate convexity, approximate monotonicity, error bound, normal cone, subconvexity, subdifferential.

MSC: 49J52, 49J53, 52A01, 52A30, 58C06, 90C26.

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