Strong and Weak Convexity of Closed Sets in a Hilbert Space
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.
KeywordsStrong 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.
- 12.M.V. Balashov, D. Repovš, Uniformly convex subsets of the Hilbert space with modulus of convexity of the second order. J. Math. Anal. Appl. 377, 754–761 (2011)Google Scholar
- 21.G. Colombo, L. Thibault, Prox-regular sets and applications, in Handbook of Nonconvex Analysis, ed. by D.Y. Gao, D. Motreanu (International Press, Somerville, MA, 2010)Google Scholar
- 24.L. Danzer, B. Grünbaum, V. Klee, Helly’s theorem and its relatives, in Convexity, ed. by V. Klee. Proceedings of Symposia in Pure Mathematics, vol. 7 (American Mathematical Society, Providence, RI, 1963), pp. 101–180Google Scholar
- 31.H. Federer, Curvature measures. Trans. Am. Math. Soc. 93, 418–491 (1959)Google Scholar
- 51.R.R. Phelps, Convex functions, monotone operators and differentiability. Lecture Notes in Mathematics, vol. 1364 (Springer, Berlin, 1989)Google Scholar
- 57.B.T. Polyak, Existence theorems and convergence of minimizing sequences in extremum problems with restrictions. Sov. Math. 7, 72–75 (1966)Google Scholar