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Proportional Subspaces of Spaces with Unconditional Basis Have Good Volume Properties

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Geometric Aspects of Functional Analysis

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

Abstract

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.

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Authors and Affiliations

  1. Mathematisches Seminar der Universität Kiel, Ludewig-Meyn-Str. 4, 24098, Kiel, Germany

    Marius Junge

Authors
  1. Marius Junge
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J. Lindenstrauss V. Milman

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© 1995 Birkhäuser Verlag Basel/Switzerland

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Junge, M. (1995). Proportional Subspaces of Spaces with Unconditional Basis Have Good Volume Properties. In: Lindenstrauss, J., Milman, V. (eds) Geometric Aspects of Functional Analysis. Operator Theory Advances and Applications, vol 77. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9090-8_12

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  • DOI: https://doi.org/10.1007/978-3-0348-9090-8_12

  • Publisher Name: Birkhäuser Basel

  • Print ISBN: 978-3-0348-9902-4

  • Online ISBN: 978-3-0348-9090-8

  • eBook Packages: Springer Book Archive

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