Advertisement

Strong and Weak Convexity of Closed Sets in a Hilbert Space

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

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 

Mathematcal Subject Classification (2010):

46C05 49J52 49J53 52A01 

Notes

Acknowledgements

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.

References

  1. 1.
    S.M. Ageev, D. Repovš, On selection theorems with decomposable values. Topol. Meth. Nonlinear Anal. 15, 385–399 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    E. Asplund, R.T. Rockafellar, Gradients of convex functions. Trans. Am. Math. Soc. 139, 443–467 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    M.V. Balashov, Proximal smoothness of a set with the Lipschitz metric projection. J. Math. Anal. Appl. 406, 360–363 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    M.V. Balashov, Antidistance and antiprojection in the Hilbert space. J. Convex Anal. 22, 521–536 (2015)MathSciNetzbMATHGoogle Scholar
  5. 5.
    M.V. Balashov, M.O. Golubev, About the Lipschitz property of the metric projection in the Hilbert space. J. Math. Anal. Appl. 394, 545–551 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    M.V. Balashov, M.O. Golubev, Weak concavity of the antidistance function. J. Convex Anal. 21, 951–964 (2014)MathSciNetzbMATHGoogle Scholar
  7. 7.
    M.V. Balashov, G.E. Ivanov, On farthest points of sets. Math. Notes 80, 163–170 (2006)MathSciNetzbMATHGoogle Scholar
  8. 8.
    M.V. Balashov, G.E. Ivanov, Properties of the metric projection on weakly vial-convex sets and parametrization of set-valued mappings with weakly convex images. Math. Notes 80, 461–467 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    M.V. Balashov, D. Repovš, On the splitting problem for selections. J. Math. Anal. Appl. 355, 277–287 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    M.V. Balashov, D. Repovš, Uniform convexity and the splitting problem for selections. J. Math. Anal. Appl. 360, 307–316 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    M.V. Balashov, D. Repovš, Weakly convex sets and modulus of nonconvexity. J. Math. Anal. Appl. 371, 113–127 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 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
  13. 13.
    F. Bernard, L. Thibault, N. Zlateva, Characterization of proximal regular sets in super reflexive Banach spaces. J. Convex Anal. 13, 525–559 (2006)MathSciNetzbMATHGoogle Scholar
  14. 14.
    F. Bernard, L. Thibault, N. Zlateva, Prox-regular sets and epigraphs in uniformly convex Banach spaces: various regularity and other properties. Trans. Am. Math. Soc. 363, 2211–2247 (2011)CrossRefzbMATHGoogle Scholar
  15. 15.
    M. Bounkhel, L. Thibault, On various notions of regularity of sets in nonsmooth analysis. Nonlinear Anal. 48, 223–246 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    A. Canino, On p-convex sets and geodesics. J. Differ. Equ. 75, 118–157 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    A. Canino, Local properties of geodesics on p-convex sets. Ann. Mat. Pura Appl. 159, 17–44 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    F.H. Clarke, R.J. Stern, P.R. Wolenski, Proximal smoothness and the lower-\(\mathcal{C}^{2}\) property. J. Convex Anal. 2, 117–144 (1995)MathSciNetzbMATHGoogle Scholar
  19. 19.
    G. Colombo, V.V. Goncharov, Variational inequalities and regularity properties of closed sets in Hilbert spaces. J. Convex Anal. 8, 197–221 (2001)MathSciNetzbMATHGoogle Scholar
  20. 20.
    G. Colombo, V.V. Goncharov, Continuous selections via geodesics. Topol. Meth. Nonlinear Anal. 18, 171–182 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 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
  22. 22.
    G. Colombo, P. Wolenski, Variational analysis for a class of minimal time functions in a Hilbert space. J. Convex Anal. 11, 335–361 (2004)MathSciNetzbMATHGoogle Scholar
  23. 23.
    G. Colombo, V.V. Goncharov, B.S. Mordukhovich, Well-posedness of minimal time problems with constant dynamics in Banach spaces. Set-Valued Var. Anal. 18, 349–372 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 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
  25. 25.
    F.S. De Blasi, J. Myjak, On a generalized best approximation problem. J. Approx. Theory 94, 54–72 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    F.S. De Blasi, G. Pianigiani, Remarks on Hausdorff continuous multifunction and selections. Comment. Math. Univ. Carol. 24, 553–561 (1983)MathSciNetzbMATHGoogle Scholar
  27. 27.
    E. De Giorgi, M. Degiovanni, A. Marino, M. Tosques, Evolution equations for a class of nonlinear operators. Atti Acad. Naz. Lincei Red. Cl. Sci. Fiz. Mat. Natur. 75, 1–8 (1983)MathSciNetzbMATHGoogle Scholar
  28. 28.
    M. Degiovanni, A. Marino, M. Tosques, Evolution equations with lack of convexity. Nonlinear Anal. 9, 1401–1443 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    M. Edelstein, On nearest points of sets in uniformly convex Banach spaces. J. Lond. Math. Soc. 43, 375–377 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    N.V. Efimov, S.B. Stechkin, Support properties of sets in Banach spaces and Chebyshev sets. Doklady Acad. Sci. USSR 127, 254–257 (1959)zbMATHGoogle Scholar
  31. 31.
    H. Federer, Curvature measures. Trans. Am. Math. Soc. 93, 418–491 (1959)Google Scholar
  32. 32.
    A. Fryszkowski, Continuous selections for a class of non-convex multivalued maps. Stud. Math. 76, 163–174 (1983)MathSciNetzbMATHGoogle Scholar
  33. 33.
    V.V. Goncharov, F.F. Pereira, Neighbourhood retractions of nonconvex sets in a Hilbert space via sublinear functionals. J. Convex Anal. 18, 1–36 (2011)MathSciNetzbMATHGoogle Scholar
  34. 34.
    V.V. Goncharov, F.F. Pereira, Geometric conditions for regularity in a time-minimum problem with constant dynamics. J. Convex Anal. 19, 631–669 (2012)MathSciNetzbMATHGoogle Scholar
  35. 35.
    V.V. Goncharov, A.A. Tolstonogov, Joint continuous selections of multivalued mappings with nonconvex values, and their applications. Math. USSR Sb. 73, 319–339 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    V.V. Goncharov, A.A. Tolstonogov, Continuous selections of the family of nonconvex-valued mappings with a noncompact domain. Sib. Math. J. 35, 479–494 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    G.E. Ivanov, Weak convexity in the senses of Vial and Efimov-Stechkin. Izv. RAN. Ser. Mat. 69, 35–60 (2005)CrossRefzbMATHGoogle Scholar
  38. 38.
    G.E. Ivanov, Weakly convex sets and their properties. Mat. Zametki 79, 60–86 (2006)MathSciNetCrossRefGoogle Scholar
  39. 39.
    G.E. Ivanov, Weakly Convex Sets and Functions: Theory and Applications (Fizmatlit, Moscow, 2006) (in Russian)zbMATHGoogle Scholar
  40. 40.
    G.E. Ivanov, Farthest points and strong convexity of sets. Math. Notes 87, 355–366 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    G.E. Ivanov, Continuity and selections of the intersection operator applied to nonconvex sets. J. Convex Anal. 22, 939–962 (2015)MathSciNetzbMATHGoogle Scholar
  42. 42.
    G.E. Ivanov, Sharp estimates for the moduli of continuity of metric projections onto weakly convex sets. Izvestiya Math. 79, 668–697 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    G.E. Ivanov, Weak convexity of sets and functions in a Banach space. J. Convex Anal. 22, 365–398 (2015)MathSciNetzbMATHGoogle Scholar
  44. 44.
    G.E. Ivanov, M.S. Lopushanski, Well-posedness of approximation and optimization problems for weakly convex sets and functions. J. Math. Sci. 209, 66–87 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    V. Klee, Circumspheres and inner products. Math. Scand. 8, 363–370 (1960)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    J. Lindenstrauss, On the modulus on smoothness and divergent series in Banach spaces. Mich. Math. J. 10, 241–252 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    E. Michael, Continuous selections. I. Ann. Math. 63, 361–382 (1956)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    E. Michael, Paraconvex sets. Math. Scand. 7, 372–376 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    J.J. Moreau, Intersection of moving convex sets in a normed space. Math. Scand. 36, 159–173 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    J.-P. Penot, Preservation of persistence and stability under intersections and operations. I. Persistence. J. Optim. Theory Appl. 79, 525–550 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    R.R. Phelps, Convex functions, monotone operators and differentiability. Lecture Notes in Mathematics, vol. 1364 (Springer, Berlin, 1989)Google Scholar
  52. 52.
    A. Pliś, Uniqueness of optimal trajectories for non-linear control systems. Ann. Polon. Math. 29, 397–401 (1975)MathSciNetzbMATHGoogle Scholar
  53. 53.
    R.A. Poliquin, R.T. Rockafellar, L. Thibault, Local differentiability of distance functions. Trans. Am. Math. Soc. 353, 5231–5249 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    E.S. Polovinkin, On strongly convex sets. Phys. J. 2, 43–59 (1996)zbMATHGoogle Scholar
  55. 55.
    E.S. Polovinkin, Strongly convex analysis. Matem. Sb. 187, 103–130 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    E.S. Polovinkin, M.V. Balashov, Elements of Convex and Strongly Convex Analysis (Fizmatlit, Moscow, 2004) (in Russian)zbMATHGoogle Scholar
  57. 57.
    B.T. Polyak, Existence theorems and convergence of minimizing sequences in extremum problems with restrictions. Sov. Math. 7, 72–75 (1966)Google Scholar
  58. 58.
    D. Repovš, P.V. Semenov, Sections of convex bodies and splitting problem for selections. J. Math. Anal. Appl. 334, 646–655 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Yu.G. Reshetnyak, On a generalization of convex surfaces. Matem. sbornik 40 (82), 381–398 (1956)MathSciNetGoogle Scholar
  60. 60.
    A. Shapiro, Existence and differentiability of metric projections in Hilbert spaces. SIAM J. Optim. 4, 130–141 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    I. Singer, Abstract Convex Analysis. Wiley Interscience Publication (Canadian Mathematical Society, New York, Toronto, 1997)zbMATHGoogle Scholar
  62. 62.
    S.B. Stechkin, Approximate properties of sets in linear normed spaces. Rev. Math. Pures Appl. 8, 5–18 (1963)MathSciNetzbMATHGoogle Scholar
  63. 63.
    J.-P. Vial, Strong and weak convexity of sets and functions. Math. Ops. Res. 8, 231–259 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    A. Weber, G. Reissig, Local characterization of strongly convex sets. J. Math. Anal. Appl. 400, 743–750 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

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

Personalised recommendations