Strong and Weak Convexity of Closed Sets in a Hilbert Space

Vladimir V. Goncharov () and Grigorii E. Ivanov ()
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Vladimir V. Goncharov: CIMA, Universidade de Évora
Grigorii E. Ivanov: Moscow Institute of Physics and Technology

A chapter in Operations Research, Engineering, and Cyber Security, 2017, pp 259-297 from Springer

Abstract: Abstract We give a brief survey of the geometrical and topological properties of two classes of closed sets in a Hilbert space, which strengthen and weaken the convexity concept, respectively. We prove equivalence of various characterizations of these sets, which are partially new while partially known in the literature but accompanied with different proofs. Along with the uniform notions dating back to Efimov, Stechkin, Vial, Clarke, Stern, Wolenski, and others we pay attention to some local and pointwise constructions, which can be interpreted through positive and negative scalar curvatures. In the final part of the paper we give several applications to geometry of Hilbert spaces, to set-valued analysis, and to time optimal control problem.

Keywords: Strong convexity; Weak convexity; Proximal smoothness; Prox-regularity; Proximal normal cone; Variational inequality; Minkowski operations; Continuous selections; Hausdorff continuity; 46C05; 49J52; 49J53; 52A01 (search for similar items in EconPapers)
Date: 2017
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Persistent link: https://EconPapers.repec.org/RePEc:spr:spochp:978-3-319-51500-7_12

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DOI: 10.1007/978-3-319-51500-7_12

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