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arXiv · 1007.0162

Weakly convex sets and modulus of nonconvexity

Maxim V. Balashov·Dušan Repovš
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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.

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BibTeXRIS

Maxim V. Balashov, Dušan Repovš. 2010-07-01. Weakly convex sets and modulus of nonconvexity. https://doi.org/10.1016/j.jmaa.2010.04.039

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