Strong and Weak Convexity of Closed Sets in a Hilbert Space

  • Vladimir V. GoncharovEmail author
  • Grigorii E. Ivanov
Part of the Springer Optimization and Its Applications book series (SOIA, volume 113)


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.


Strong convexity Weak convexity Proximal smoothness Prox-regularity Proximal normal cone Variational inequality Minkowski operations Continuous selections Hausdorff continuity 

Mathematcal Subject Classification (2010):

46C05 49J52 49J53 52A01 



Vladimir V. Goncharov financially supported by National Funds of Portugal through FCT—Fundação para a Ciência e a Tecnologia in the framework of the Project “UID/Mat/04674/2013 (CIMA).” Grigorii E. Ivanov supported by the Russian Foundation for basic research, project 16-01-00259-a.


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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.CIMA, Universidade de ÉvoraÉvoraPortugal
  2. 2.Institute of Systems Dynamics and Control Theory of Siberian Branch of RASIrkutskRussia
  3. 3.Moscow Institute of Physics and TechnologyMoscow RegionRussia

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