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Criteria for Convexity of Closed Sets in Banach Spaces

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Abstract

Criteria for the convexity of closed sets in general Banach spaces in terms of the Clarke and Bouligand tangent cones are proved. In the case of uniformly convex spaces, these convexity criteria are stated in terms of proximal normal cones. These criteria are used to derive sufficient conditions for the convexity of the images of convex sets under nonlinear mappings and multifunctions.

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Acknowledgments

The author is grateful to Sergey Aseev and Francis Clarke for useful discussions.

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Authors and Affiliations

  1. Department of Mathematics, Western Michigan University, Kalamazoo, MI, 49008-5248, USA

    Yuri S. Ledyaev

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  1. Yuri S. Ledyaev
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Correspondence to Yuri S. Ledyaev.

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Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2019, Vol. 304, pp. 205–220.

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Ledyaev, Y.S. Criteria for Convexity of Closed Sets in Banach Spaces. Proc. Steklov Inst. Math. 304, 190–204 (2019). https://doi.org/10.1134/S0081543819010139

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