Weakly convex sets and modulus of nonconvexity
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.
Cite
@article{arxiv.1007.0162,
title = {Weakly convex sets and modulus of nonconvexity},
author = {Maxim V. Balashov and Dušan Repovš},
journal= {arXiv preprint arXiv:1007.0162},
year = {2010}
}