Skip to main content
Log in

Modulus of Strong Proximinality and Continuity of Metric Projection

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

Abstract

In this paper we initiate a quantitative study of strong proximinality. We define a quantity ϵ(x, t) which we call as modulus of strong proximinality and show that the metric projection onto a strongly proximinal subspace Y of a Banach space X is continuous at x if and only if ϵ(x, t) is continuous at x whenever t > 0. The best possible estimate of ϵ(x, t) characterizes spaces with \(1 \frac{1}{2}\) ball property. Estimates of ϵ(x, t) are obtained for subspaces of uniformly convex spaces and of strongly proximinal subspaces of finite codimension in C(K).

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.

Institutional subscriptions

Similar content being viewed by others

References

  1. Dutta, S., Narayana, D.: Strongly proximinal subspaces in Banach spaces. Contemp. Math. - Am. Math. Soc. 435, 143–152 (2007)

    MathSciNet  Google Scholar 

  2. Dutta, S., Narayana, D.: Strongly proximinal subspaces of finite codimension in C(K). Colloq. Math. 109(1), 119–128 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Fonf, V., Lindenstrauss, J.: On the metric projection in a polyhedral space. (unpublished note)

  4. Godefroy, G., Indumathi, V.: Strong proximinality and polyhedral spaces. Rev. Mat. Complut. 14(1), 105–125 (2001)

    MathSciNet  MATH  Google Scholar 

  5. Godini, G.: Best approximation and intersection of balls. In: Banach Space Theory and its Applications. Lecture Notes in Mathematics. Springer (1981)

  6. Gleit, A., McGnigan, R.: A note on polyhedral Banach spaces. Proc. Am. Math. Soc. 33, 398–404 (1972)

    Article  MATH  Google Scholar 

  7. Indumathi, V.: Metric projections of closed subspaces of c 0 onto subspaces of finite codimension. Collq. Math. 99(2), 231–252 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Indumathi, V.: Metric projections and polyhedral spaces. Set-Valued Anal. 15, 239–250 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Harmand, H., Werner, D., Werner, W.: M-ideals in Banach spaces and Banach algebras. In: Lecture Notes in Mathematics, vol. 1547. Springer, Berlin (1993)

    Google Scholar 

  10. Lau, K.-S.: On a sufficient condition for proximity. Trans. Am. Math. Soc. 251, 343–356 (1979)

    Article  MATH  Google Scholar 

  11. Vlasov, L.P.: Approximative properties of subspaces of finite codimension in C(Q). Mat. Zametki 28(2), 205–222 (1980); English translation in: Math. Notes 28(2), 565–574 (1980)

    MathSciNet  MATH  Google Scholar 

  12. Yost, D.: The n-ball properties in real and complex Banach spaces. Math. Scand. 50(1), 100–110 (1982)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to S. Dutta.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dutta, S., Shunmugaraj, P. Modulus of Strong Proximinality and Continuity of Metric Projection. Set-Valued Anal 19, 271–281 (2011). https://doi.org/10.1007/s11228-010-0143-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11228-010-0143-y

Keywords

Mathematics Subject Classifications (2010)

Navigation