Weakly convex sets and modulus of nonconvexity

TL;DR

This paper introduces a generalized concept of weakly convex sets in Banach spaces with a modulus of nonconvexity, proving new theorems on retraction, continuity, and set intersections, and exploring their relation to proximally smooth sets.

Contribution

It extends the notion of weakly convex sets to Banach spaces with second-order convexity modulus, establishing new theorems and relationships in nonconvex analysis.

Findings

Proved a new retraction theorem for weakly convex sets.

Established continuity results for intersections of set-valued mappings.

Provided solutions to the splitting problem for selections.

Abstract

We consider a definition of a weakly convex set which is a generalization of the notion of a weakly convex set in the sense of Vial and a proximally smooth set in the sense of Clarke, from the case of the Hilbert space to a class of Banach spaces with the modulus of convexity of the second order. Using the new definition of the weakly convex set with the given modulus of nonconvexity we prove a new retraction theorem and we obtain new results about continuity of the intersection of two continuous set-valued mappings (one of which has nonconvex images) and new affirmative solutions of the splitting problem for selections. We also investigate relationship between the new definition and the definition of a proximally smooth set and a smooth set.