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Strong and Weak Convexity of Closed Sets in a Hilbert Space

In: Operations Research, Engineering, and Cyber Security

Author

Listed:
  • Vladimir V. Goncharov

    (CIMA, Universidade de Évora
    Institute of Systems Dynamics and Control Theory of Siberian Branch of RAS)

  • Grigorii E. Ivanov

    (Moscow Institute of Physics and Technology)

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.

Suggested Citation

  • Vladimir V. Goncharov & Grigorii E. Ivanov, 2017. "Strong and Weak Convexity of Closed Sets in a Hilbert Space," Springer Optimization and Its Applications, in: Nicholas J. Daras & Themistocles M. Rassias (ed.), Operations Research, Engineering, and Cyber Security, pages 259-297, Springer.
    • Handle: RePEc:spr:spochp:978-3-319-51500-7_12
    DOI: 10.1007/978-3-319-51500-7_12
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