A Note on the Banach-Mazur Distance to the Cube

  • A. A. Giannopoulos
Part of the Operator Theory Advances and Applications book series (OT, volume 77)


If X is an n-dimensional normed space, and d denotes the Banach-Mazur distance, then d(X, ℓ n ) ≤ cn 5/6.


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  1. [B-S]
    J. Bourgain, S.J. Szarek, The Banach-Mazur distance to the cube and the Dvoretzky-Rogers factorization. Israel J. Math. 62 (1988), 169–180.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [G]
    A.A. Giannopoulos, On the Banach-Mazur distance to the cube, preprint, December 1992.Google Scholar
  3. [J]
    F. John, Extremum problems with inequalities as subsidiary conditions, Courant Anniversary Volume, New York: Interscience, 1948.Google Scholar
  4. [M-Sc]
    V.D. Milman, G. Schechtman, Asymptotic theory of finite dimensional normed spaces, Springer-Verlag, Lecture Notes in Mathematics 1200 (1986).zbMATHGoogle Scholar
  5. [P]
    A. Pelczynski, Structural theory of Banach spaces and its interplay with analysis and probability, Proceedings of the ICM 1983, PWN-North Holland (1984), 237–269.Google Scholar
  6. [S]
    N. Sauer, On the density of families of sets, J. Comb. Theory, Ser. A 13 (1972), 145–147.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [Sh]
    S. Shelah, A combinatorial problem: stability and order for models and theories in infinitary languages, Pacific J. Math. 41 (1972), 247–261.MathSciNetzbMATHGoogle Scholar
  8. [S-T]
    S.J. Szarek, M. Talagrand, An isomorphic version of the Sauer-Shelah lemma and the Banach-Mazur distance to the cube. GAFA Seminar’87–’88, Springer Lecture Notes in Mathematics 1376 (1989), 105–112.Google Scholar
  9. [Szl]
    S.J. Szarek, Spaces with large distance to ℓn and random matrices, American J. Math. 112 (1990), 899–942.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [Sz2]
    S.J. Szarek, On the geometry of the Banach-Mazur compactum, Lecture Notes in Mathematics 1470 (1991), 48–59.Google Scholar
  11. [Sz3]
    S.J. Szarek, Personal communication.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 1995

Authors and Affiliations

  • A. A. Giannopoulos
    • 1
  1. 1.Department of MathematicsCase Western Reserve UniversityClevelandUSA

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