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The Banach-Mazur distance to the cube and the Dvoretzky-Rogers factorization

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Abstract

We show the existence of a sequence (λ n ) of scalars withλ n =o(n) such that, for any symmetric compact convex bodyB ⊂R n, there is an affine transformationT satisfyingQ ⊂T(B) ⊂λ n Q, whereQ is then-dimensional cube. This complements results of the second-named author regarding the lower bound on suchλ n . We also show that ifX is ann-dimensional Banach space andm=[n/2], then there are operatorsα:l m2 →X andβ:X→l m∞ with ‖α‖·‖β‖≦C, whereC is a universal constant; this may be called “the proportional Dvoretzky-Rogers factorization”. These facts and their corollaries reveal new features of the structure of the Banach-Mazur compactum.

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Authors and Affiliations

  1. Institut des Hautes Etudes Scientifiques, 35 Route de Chartres, 91440, Bures-sur-Yvette, France

    J. Bourgain

  2. Case Western Reserve University, 44106, Cleveland, OH, USA

    S. J. Szarek

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  1. J. Bourgain
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  2. S. J. Szarek
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Research performed while this author was visiting IHES. Supported in part by the NSF Grant DMS-8702058 and the Sloan Research Fellowship.

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Bourgain, J., Szarek, S.J. The Banach-Mazur distance to the cube and the Dvoretzky-Rogers factorization. Israel J. Math. 62, 169–180 (1988). https://doi.org/10.1007/BF02787120

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  • DOI: https://doi.org/10.1007/BF02787120

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