Abstract

We present a sufficient condition for the uniform continuity of metric projection. This condition is a natural strengthening of the notion of strong proximinality, appearing in the literature of the past few years. We show that this condition is equivalent to the sufficient condition of continuity of metric projection introduced by K.-S. Lau back in 1979. A characterization of uniform convexity through proximinality is presented and we also relate quantitatively the power type estimate of modulus of uniform strong proximinality to the power type estimate of modulus of uniform convexity.

Author information

Contact details are reproduced from the original publication and may be historical.

Sudipta Dutta

Dept. of Mathematics and Statistics, Indian Institute of Technology, Kanpur 208016, India

sudipta@iitk.ac.in

P. Shunmugaraj

Dept. of Mathematics and Statistics, Indian Institute of Technology, Kanpur 208016, India

psraj@iitk.ac.in

Vamsinadh Thota

Dept. of Mathematics and Statistics, Indian Institute of Technology, Kanpur 208016, India

vamst@iitk.ac.in

Keywords

  • Uniformly strongly proximinal
  • U-proximinal
  • metric projection
  • uniform convexity

Mathematics Subject Classification

41A65, 46B20

References

  1. A. K. Chakrabarty, P. Shunmugaraj, C. Zălinescu: Continuity properties for the subdifferential and ε-subdifferential of a convex function and its conjugate, J. Convex Analysis 14(3) (2007) 479–514.https://doi.org/10.68381/jca14031Google Scholar
  2. F. Deutsch, P. Kenderov: Continuous selections and approximate selection for set-valued mappings and applications to metric projections, SIAM J. Math. Anal. 14(1) (1983) 185–194.https://doi.org/10.1137/0514015Google Scholar
  3. S. Dutta, D. Narayana: Strongly proximinal subspaces in Banach spaces, Contemp. Math., Amer. Math. Soc. 435 (2007) 143–152.https://doi.org/10.1090/conm/435/08372Google Scholar
  4. S. Dutta, D. Narayana: Strongly proximinal subspaces of finite codimension in C(K), Colloq. Math. 109(1) (2007) 119–128.https://doi.org/10.4064/cm109-1-10Google Scholar
  5. S. Dutta, P. Shunmugaraj: Modulus of strong proximinality and continuity of metric projection, Set-Valued Var. Analysis 19(2) (2011) 271–281.https://doi.org/10.1007/s11228-010-0143-yGoogle Scholar
  6. S. Dutta, P. Shunmugaraj: Strong proximinality of closed convex sets, J. Approx. Theory 163(4) (2011) 547–553.https://doi.org/10.1016/j.jat.2011.01.001Google Scholar
  7. V. Fonf, J. Lindenstrauss, L. Vesely: Best approximation in polyhedral Banach spaces, J. Approx. Theory 163(11) (2011) 1748–1771.https://doi.org/10.1016/j.jat.2011.06.011Google Scholar
  8. C. Franchetti, R. Payá: Banach spaces with strongly subdifferentiable norm, Boll. Un. Mat. Ital. B 7(1) (1993) 45–70.Google Scholar
  9. G. Godefroy, V. Indumathi: Strong proximinality and polyhedral spaces, Rev. Mat. Complut. 14(1) (2001) 105–125.https://doi.org/10.5209/rev_REMA.2001.v14.n1.17047Google Scholar
  10. G. Godini: Best approximation and intersections of balls, in: Banach Space Theory and its Applications, Lecture Notes in Mathematics 991, Springer-Verlag, Berlin (1983) 44–54.https://doi.org/10.1007/BFb0061557Google Scholar
  11. P. Harmand, D. Werner, W. Werner: M-ideals in Banach Spaces and Banach Algebras, Lecture Notes in Mathematics 1547, Springer-Verlag, Berlin (1993).https://doi.org/10.1007/BFb0084355Google Scholar
  12. R. Holmes: A course on optimization and best approximation, Lecture Notes in Mathematics 257, Springer-Verlag, Berlin-New York (1972).https://doi.org/10.1007/BFb0059450Google Scholar
  13. R. Holmes: M-ideals in Approximation Theory. Approximation theory II, Academic Press, New York (1976).Google Scholar
  14. V. Indumathi: Metric projections of closed subspaces of c0 onto subspaces of finite codimension, Colloq. Math. 99(2) (2004) 231–252.https://doi.org/10.4064/cm99-2-8Google Scholar
  15. V. Indumathi: Metric projections and polyhedral spaces, Set-Valued Anal. 15(3) (2007) 239–250.https://doi.org/10.1007/s11228-006-0036-2Google Scholar
  16. B. Kripke, R. Holmes: Approximation of bounded functions by continuous functions, Bull. Amer. Math. Soc. 71 (1965) 896–897.https://doi.org/10.1090/S0002-9904-1965-11433-1Google Scholar
  17. K.-S. Lau: On a sufficient condition for proximity, Trans. Amer. Math. Soc. 251 (1979) 343–356.https://doi.org/10.1090/S0002-9947-1979-0531983-0Google Scholar
  18. R. E. Megginson: An Introduction to Banach Space Theory, Springer-Verlag, New York (1998).https://doi.org/10.1007/978-1-4612-0603-3Google Scholar
  19. J. B. Prolla, A. O. Chiacchio: Proximinality of certain subspaces of Cb (S; E), Rocky Mountain J. Math. 19(1) (1989) 335–344.https://doi.org/10.1216/RMJ-1989-19-1-335Google Scholar
  20. T. S. S. R. K. Rao: The one and half ball property in spaces of vector-valued functions, J. Convex Analysis 20(1) (2013) 13–23.https://doi.org/10.68381/jca20002Google Scholar
  21. D. Yost: Approximation by compact operators between C(X) spaces, J. Approx. Theory 49(2) (1987) 99–109.https://doi.org/10.1016/0021-9045(87)90080-3Google Scholar

Suggested citation

S. Dutta, P. Shunmugaraj, V. Thota. “Uniform Strong Proximinality and Continuity of Metric Projection.” Journal of Convex Analysis 24 (2017), No. 4, 1263–1279. https://doi.org/10.68381/jca24072

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Published by Heldermann Verlag, 2017. Rights now held by Banach Press.