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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 139597, 9 pages
http://dx.doi.org/10.1155/2011/139597
Research Article

Convexities and Existence of the Farthest Point

College of Fundamental Studies, Shanghai University of Engineering Science, Shanghai 201600, China

Received 10 July 2011; Revised 20 September 2011; Accepted 20 September 2011

Academic Editor: Toka Diagana

Copyright © 2011 Z. H. Zhang and C. Y. Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. E. Asplund, “Farthest points in reflexive locally uniformly rotund Banach spaces,” Israel Journal of Mathematics, vol. 4, no. 4, pp. 213–216, 1966. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. M. Edelstein, “Farthest points of sets in uniformly convex Banach spaces,” Israel Journal of Mathematics, vol. 4, no. 3, pp. 171–176, 1966. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. S. Elumalai and R. Vijayaraghavan, “Farthest points in normed linear spaces,” Journal General Mathematics, vol. 40, no. 3, pp. 9–22, 2006. View at Google Scholar
  4. K. S. Lau, “Farthest points in weakly compact sets,” Israel Journal of Mathematics, vol. 22, no. 2, pp. 168–174, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. E. Naraghirad, “Characterizations of simultaneous farthest point in normed linear spaces with applications,” Optimization Letters, vol. 3, no. 1, pp. 89–100, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. Z. H. Zhang and Z. R. Shi, “Convexities and approximative compactness and continuity of metric projection in Banach spaces,” Journal of Approximation Theory, vol. 161, no. 2, pp. 802–812, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. P. Bandyopadhyay, D. Huang, B.-L. Lin, and S. L. Troyanski, “Some generalizations of locally uniform rotundity,” Journal of Mathematical Analysis and Applications, vol. 252, no. 2, pp. 906–916, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. Z. H. Zhang and C. Y. Liu, “Some generalizations of locally and weakly locally uniformly convex space,” Nonlinear Analysis, Theory, Methods and Applications, vol. 74, no. 12, pp. 3896–3902, 2011. View at Publisher · View at Google Scholar
  9. F. Sullivan, “Geometrical peoperties determined by the higher duals of a Banach space,” Illinois Journal of Mathematics, vol. 21, no. 2, pp. 315–318, 1977. View at Google Scholar
  10. Z. H. Zhang and C. J. Zhang, “On very rotund Banach space,” Applied Mathematics and Mechanics, vol. 21, no. 8, pp. 965–970, 2000 (Chinese). View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. H. J. Wang and Z. H. Zhang, “Characterizations of property (CK),” Acta Mathematica Scientia. A, vol. 17, no. 3, pp. 280–284, 1997 (Chinese). View at Google Scholar · View at Zentralblatt MATH
  12. C. X. Wu and Y. J. Li, “Strong convexity in Banach spaces,” Journal of Mathematics, vol. 13, no. 1, pp. 105–108, 1993. View at Google Scholar · View at Zentralblatt MATH
  13. R. B. Holmes, Geometric Functional Analysis and Its Applications, Springer, New York, NY, USA, 1975.
  14. M. A. Smith, “A Banach space that is MLUR but not HR,” Mathematische Annalen, vol. 256, no. 2, pp. 277–279, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. V. I. Isratescu, Strict Convexity and Complex Strict Convex: Theory and Applied Mathematics, vol. 89 of Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1984.
  16. B. L. Lin and X. T. Yu, “On K-uniform rotund and the fully convex Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 110, no. 2, pp. 407–410, 1985. View at Publisher · View at Google Scholar · View at MathSciNet
  17. M. A. Smith, “Some examples concerning rotundity in Banach spaces,” Mathematische Annalen, vol. 233, no. 2, pp. 155–161, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. R. Ni and C. Li, “On well posedness of farthest and simultaneous farthest point problems in Banach spaces,” Acta Mathematica Sinica, vol. 43, no. 3, pp. 421–426, 2000. View at Google Scholar · View at Zentralblatt MATH