Proportional Subspaces of Spaces with Unconditional Basis Have Good Volume Properties

  • Marius Junge
Part of the Operator Theory Advances and Applications book series (OT, volume 77)


A generalization of Lozanovskii’s result is proved. Let E be k-dimensional sub-space of an n-dimensional Banach space with unconditional basis. Then there exist x 1,…, x k E such that B E absconv{x 1,…, x k } and
$$ {\left( {\frac{{vol\left( {absconv\left\{ {{x_1}, \ldots {x_k}} \right\}} \right)}} {{vol\left( {{B_E}} \right)}}} \right)^{\frac{1}{k}}} \leqslant {\left( {e\frac{n}{k}} \right)^2} $$
This answers a question of V. Milman which appeared during a GAFA seminar talk about the hyperplane problem. We add logarithmical estimates concerning the hyperplane conjecture for proportional subspaces and quotients of Banach spaces with unconditional basis.


Banach Space Unit Ball Convex Body Isometric Embedding Unconditional Basis 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [BA]
    K. M. Ball: Normed spaces with a weak-Gordon-Lewis property; Proc. of Funct. Anal., University of Texas and Austin 1987–1989, Springer Lect. Notes 1470, 36–47.Google Scholar
  2. [BAPA]
    K. Ball and A. Pajor: The entropy of convex bodies with “few” faces; Geometry of Banach spaces (Strobl 1989) 25–32, London Math. Soc., Lect. Notes Ser 158.Google Scholar
  3. [BM]
    J. Bourgain and V. D. Milman: New volume ratio properties for convex symmetric bodies inn; Inv. Math. 88 (9187), 319–340.Google Scholar
  4. [FIJ]
    T. Figiel and W.B. Johnson: Large subspaces of n and estimates of the Gordon-Lewis constants, Isr. J. of Math. 37 (1980), 92–112.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [GL]
    E. Gluskin; Extremal properties of rectangular parallelepipeds and their application to Banach spaces; Math. Sbornik 136 (178)(1988), 85–95.Google Scholar
  6. [GOS]
    Y. Gordon and P. Saphar: Ideal norms on E ⊗L P, Illinois J. of Math. 21 (1979), 266–285.MathSciNetGoogle Scholar
  7. [HEN]
    D. Hensley: Slicing convex bodies-bounds of slice area in terms of the body’s covariance; Proc. of AMS 79 (1980), 619–625.MathSciNetzbMATHGoogle Scholar
  8. [JU]
    M. Junge: Hyperplane conjecture for spaces of p; Forum Math. 6 (1994), 617–635.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [LO]
    G. Ya. Lozanovskii: On some Banach lattices; Siberian Math. J. 10 (1969), 419–431.CrossRefGoogle Scholar
  10. [MCM]
    P. McMullen: Volume of projections of unit cubes; Bull. London Math. Soc. 16 (1984), 278–280.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [MEP]
    M. Meyer and A. Pajor: Sections of the unit ball of pn; J. of Funct. Anal. 80 (1988), 109–123.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [MIPA]
    V. D. Milman and A. Pajor: Isotropic position, inertia ellipsoids and zonoid of the unit ball of a normed n-dimensional space; GAFA Seminar’87–89, Springer Lect. Notes in Math. 1376 (1989), 64–104.Google Scholar
  13. [MIS]
    V. D. Milman and G. Schechtman: Asymptotic theory of finite dimensional normed spaces; Springer Lect. Notes in Math. 1200 (1986).Google Scholar
  14. [RE]
    S. Reisner: Random Polytopes and the volume product of symmetric convex bodies; Math. Scand. 57 (1985), 386–392.MathSciNetzbMATHGoogle Scholar
  15. [PS]
    G. Pisier: Factorization of linear operators and Geometry of Banach spaces; CBMS Regional Conference Series n ° 60, AMS 1986.Google Scholar
  16. [PI1]
    A. Pietsch: Operator ideals; VEB Berlin 1979 and North Holland 1980.Google Scholar
  17. [PI2]
    A. Pietsch: Operator Ideals; Deutscher Verlag Wiss., Berlin 1978 and North Holland, Amsterdam — New York — Oxford 1980. Cambridge University Press, 1987.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 1995

Authors and Affiliations

  • Marius Junge
    • 1
  1. 1.Mathematisches Seminar der Universität KielKielGermany

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