Skip to main content
Log in

Nearly Chebyshev Sets are Almost Convex

  • Published:
Set-Valued and Variational Analysis Aims and scope Submit manuscript

Abstract

In this paper we show that nearly Chebyshev sets are almost convex and nearly uniquely remotal sets are almost singleton sets.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Asplund, E.: Cebyšev sets in Hilbert spaces. Trans. Amer. Math. Soc. 144, 235–240 (1969)

    MathSciNet  MATH  Google Scholar 

  2. Balaganskii, V.S., Vlasov, L.P.: The problem of the convexity of Chebyshev sets. Russian Math. Surveys 51(6), 1127–1190 (1996)

    Article  MathSciNet  Google Scholar 

  3. Borwein, J.M.: Proximality and Chebyshev sets. Optim. Lett. 1, 21–32 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Borwein, J.M., Fitzpatrick, S.: Existence of nearest points in Banach spaces. Canad. J Math. 41, 702–720 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  5. Borwein, J.M., Lewis, A.S.: Convex Analysis and Nonlinear Optimization. Theory and Examples, Second edition. CMS Springer, New York (2006)

    Book  MATH  Google Scholar 

  6. Borwein, J.M., Preiss, D.: A smooth variational principle with applications to subdifferentiability and differentiability of convex functions. Trans. Amer. Math. Soc. 303, 517–527 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  7. Brøndsted, A., Rockafellar, R.T.: On the subdifferentiability of convex functions. Proc. Amer. Math. Soc. 16, 605–611 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  8. Efimov, N.V., Steckin, S.B.: Approximative compactness and Chebyshev sets (in Russian). Dokl. Akad. Nauk SSSR 140, 522–524 (1961)

    MathSciNet  Google Scholar 

  9. Ekeland, I.: Nonconvex minimization problems. Bull. Amer. Math. Soc. (N.S.) 1(3), 443–474 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fan, K.y.: Fixed-point and minimax theorems in locally convex topological linear spaces. Proc. Nat. Acad. Sci. U. S. A. 38, 121–126 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fletcher, J., Moors, W.B.: Chebyshev sets. J. Austral. Math. Soc. 98, 161–231 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Glicksberg, I.L.A.: A further generalization of the Kakutani fixed theorem, with application to Nash equilibrium points. Proc. Amer. Math. Soc. 3, 170–174 (1952)

    MathSciNet  MATH  Google Scholar 

  13. Jiang, M.: On Johnson’s example of a nonconvex Chebyshev set. J. Approx. Theory 74(2), 152–158 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  14. Johnson, G.G.: A nonconvex set which has the unique nearest point property. J. Approx. Theory 51(4), 289–332 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kenderov, P.S., Moors, W.B., Revalski, J.P.: Dense continuity and selections of set-valued mappings. Serdica Math. J. 24, 49–72 (1998)

    MathSciNet  MATH  Google Scholar 

  16. Moors, W.B.: A selection theorem for weak upper semi-continuous set-valued mappings. Bull. Austral. Math. Soc. 53, 213–227 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  17. Phelps, R.R.: Convex Functions, Monotone Operators and Differentiability. Lecture Notes in Mathematics. Springer, Berlin (1993)

    Google Scholar 

  18. Riesz, F.: Sur une espece de géométrie analytique des systémes de fonctions sommables. C. R. Acad. Sci. Paris 144, 1409–1411 (1907)

    MATH  Google Scholar 

  19. Royden, H.L.: Real Analysis, Third Edition. Macmillan Publishing Company, New York (1988)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Warren B. Moors.

Additional information

Dedicated to the memory of Jonathan M. Borwein, amentor and adear friend.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Moors, W.B. Nearly Chebyshev Sets are Almost Convex. Set-Valued Var. Anal 26, 67–76 (2018). https://doi.org/10.1007/s11228-017-0445-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11228-017-0445-4

Keywords

Mathematics Subject Classification (2010)

Navigation