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A pythagorean approach in Banach spaces


Let be a Banach space and let be the unit sphere of. Parameters,,, and, where and are introduced and studied. The values of these parameters in the spaces and function spaces are estimated. Among the other results, we proved that a Banach space with, or is uniform nonsquare; and a Banach space with, or has uniform normal structure.



  1. Brodskiĭ MS, Mil'man DP: On the center of a convex set. Doklady Akademii Nauk. SSSR (N.S.) 1948, 59: 837–840.

    MathSciNet  Google Scholar 

  2. Busemann H: The Geometry of Geodes. Academic Press, New York; 1955:x+422.

    Google Scholar 

  3. Clarkson JA: Uniformly convex spaces. Transactions of the American Mathematical Society 1936,40(3):396–414. 10.1090/S0002-9947-1936-1501880-4

    Article  MathSciNet  MATH  Google Scholar 

  4. Day MM: Normed Linear Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. Volume 21. 3rd edition. Springer, New York; 1973:viii+211.

    Google Scholar 

  5. Gao J: Normal hexagon and more general Banach spaces with uniform normal structure. Journal of Mathematics. Shuxue Zazhi 2000,20(3):241–248.

    MATH  MathSciNet  Google Scholar 

  6. Gao J: Normal structure, fixed points and related parameters in Banach spaces. Journal of Dynamical Systems and Geometric Theories 2002,1(1):1–18.

    Article  MATH  MathSciNet  Google Scholar 

  7. Gao J, Lau K-S: On two classes of Banach spaces with uniform normal structure. Polska Akademia Nauk. Instytut Matematyczny. Studia Mathematica 1991,99(1):41–56.

    MATH  MathSciNet  Google Scholar 

  8. García-Falset J: Stability and fixed points for nonexpansive mappings. Houston Journal of Mathematics 1994,20(3):495–506.

    MATH  MathSciNet  Google Scholar 

  9. James RC: Uniformly non-square Banach spaces. Annals of Mathematics. Second Series (2) 1964, 80: 542–550. 10.2307/1970663

    Article  MATH  MathSciNet  Google Scholar 

  10. Kirk WA: A fixed point theorem for mappings which do not increase distances. The American Mathematical Monthly 1965, 72: 1004–1006. 10.2307/2313345

    Article  MATH  MathSciNet  Google Scholar 

  11. Mazcuñán-Navarro EM: On the modulus of-convexity of Ji Gao. Abstract and Applied Analysis 2003,2003(1):49–54. 10.1155/S1085337503204127

    Article  MATH  MathSciNet  Google Scholar 

  12. Schäffer JJ: Geometry of Spheres in Normed Spaces, Lecture Notes in Pure and Applied Mathematics, no. 20. Marcel Dekker, New York; 1976:vi+228.

    Google Scholar 

  13. Sims B: "Ultra"-Techniques in Banach Space Theory, Queen's Papers in Pure and Applied Mathematics. Volume 60. Queen's University, Ontario; 1982:iv+117.

    Google Scholar 

  14. Sims B: A class of spaces with weak normal structure. Bulletin of the Australian Mathematical Society 1994,49(3):523–528. 10.1017/S0004972700016634

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to Ji Gao.

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Gao, J. A pythagorean approach in Banach spaces. J Inequal Appl 2006, 94982 (2006).

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