Skip to main content
SpringerLink
Log in

Proximality and Chebyshev sets

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

This paper is a companion to a lecture given at the Prague Spring School in Analysis in April 2006. It highlights four distinct variational methods of proving that a finite dimensional Chebyshev set is convex and hopes to inspire renewed work on the open question of whether every Chebyshev set in Hilbert space is convex.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Asplund E. (1969) Cebysev sets in Hilbert space. Trans. Am. Math. Soc. 144, 235–240

    Article  MATH  MathSciNet  Google Scholar 

  2. Borwein J.M. (1984) Integral characterizations of Euclidean space. Bull. Aust. Math. Soc. 29, 357–364

    Article  MATH  Google Scholar 

  3. Borwein J.M.: David Bailey: Mathematics by Experiment. A K Peters Ltd., Natick (2004)

  4. Borwein J.M., Fitzpatrick S.P. (1989) Existence of nearest points in Banach spaces. Can. J. Math. 61, 702–720

    MathSciNet  Google Scholar 

  5. Borwein J.M., Fitzpatrick S.P., Giles J.R. (1987) The differentiability of real functions on normed linear spaces using generalized gradients. J. Optim. Theory Appl. 128, 512–534

    MATH  MathSciNet  Google Scholar 

  6. Borwein J.M., Giles J.R. (1987) The proximal normal formula in Banach space. Trans. Am. Math. Soc. 302:371–381

    Article  MATH  MathSciNet  Google Scholar 

  7. Borwein J.M., Lewis A.S.: Convex Analysis and Nonlinear Optimization, (enlarged edition). CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC. Springer, Berlin Heidelberg New York (2005)

  8. Borwein J.M., Treiman J.S., Zhu Q.J. (1999) Partially smooth variational principles and applications. Nonlinear Anal. 35, 1031–1059

    Article  MathSciNet  Google Scholar 

  9. Borwein J.M., Vanderwerff J.S.: Convex Functions: A Handbook. Encyclopedia of Mathematics and Applications. Cambridge University Press, Cambridge (2007) (to appear)

  10. Borwein J.M., Zhu Q.J. (2005) Techniques of Variational Analysis: an Introduction CMS Books in Mathematics. Springer, Berlin Heidelberg New York

    Google Scholar 

  11. Brown A.L. (1974) A rotund, reflexive space having a subspace of co-dimension 2 with a discontinuous metric projection. Michigan Math. J. 21, 145–151

    Article  MATH  MathSciNet  Google Scholar 

  12. Clarke F.H. (1983) Optimization and Nonsmooth Analysis Canadian Mathematical Society Series of Monographs and Advanced Texts. Wiley, New York

    Google Scholar 

  13. Deutsch F.R. (2001) Best Approximation in Inner Product Spaces. Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

  14. Duda J. (2006) On the size of the set of points where the metric projection is discontinuous. J. Nonlinear Convex Anal. 7, 67–70

    MATH  MathSciNet  Google Scholar 

  15. Fabian M., Habala P., Hájek P., Montesinos V., Pelant J., Zizler V. (2001) Functional Analysis and Infinite-dimensional Geometry CMS Books in Mathematics. Springer, Berlin Heidelberg New York

    Google Scholar 

  16. Giles J.R. (1982) Convex Analysis with Application in the Differentiation of Convex Functions. Pitman, Boston

    MATH  Google Scholar 

  17. Giles J.R.(1988) Differentiability of distance functions and a proximinal property inducing convexity. Proc. Am. Math. Soc. 104, 458–464

    Article  MATH  MathSciNet  Google Scholar 

  18. Hiriart-Urruty J.-B. (1983) A short proof of the variational principle for approximate solutions of a minimization problem. Am. Math. Monthly 90, 206–207

    Article  MATH  MathSciNet  Google Scholar 

  19. Hiriart-Urruty J.-B. (1998) Ensemble de Tchebychev vs ensemble convexe: l’état de la situation vu via l’analyse convexe nonlisse. Ann. Sci. Math. Québec 22, 47–62

    MATH  MathSciNet  Google Scholar 

  20. Hiriart-Urruty J.-B., Lemaréchal C. (1993) Convex Analysis and Minimization Algorithms. Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

  21. Johnson Gordon G. (2005) Closure in a Hilbert space of a preHilbert space Chebyshev set. Topology Appl. 153, 239–244

    Article  MathSciNet  MATH  Google Scholar 

  22. Klee V. (1961) Convexity of Cebysev sets. Math. Ann. 142, 291–304

    Article  MathSciNet  Google Scholar 

  23. Koščeev V.A. (1979) An example of a disconnected sun in a Banach space. (Russian). Mat. Zametki 158, 89–92

    Google Scholar 

  24. Lau K.S. (1978) Almost Chebyshev subsets in reflexive Banach spaces. Indiana Univ. Math. J 2, 791–795

    Article  Google Scholar 

  25. Rockafellar R.T. (1970) Convex Analysis. Princeton University Press, Princeton

    MATH  Google Scholar 

  26. Vlasov L.P. (1970) Almost convex and Chebyshev subsets. Math. Notes Acad. Sci. USSR 8, 776–779

    Article  MATH  MathSciNet  Google Scholar 

  27. Westfall J., Frerking U. (1989) On a property of metric projections onto closed subsets of Hilbert spaces. Proc. Am. Math. Soc. 105, 644–651

    Article  Google Scholar 

  28. Zălinescu C. (2002) Convex Analysis in General Vector Spaces. World Scientific Press, Singapore

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Computer Science, Dalhousie University, Halifax, NS, Canada

    Jonathan M. Borwein

Authors
  1. Jonathan M. Borwein
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jonathan M. Borwein.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Borwein, J.M. Proximality and Chebyshev sets. Optimization Letters 1, 21–32 (2007). https://doi.org/10.1007/s11590-006-0014-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-006-0014-5

Keywords

1991 Mathematics Subject Classification

Search

Navigation