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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)

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.

Keywords

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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Copyright information

© Birkhäuser Verlag Basel/Switzerland 1995

Authors and Affiliations

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

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