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Comptes Rendus Mathématique
Volume 335, n° 4
pages 345-350 (2002)
Doi : S1631-073X(02)02476-7
Received : 7 May 2002 ;  accepted : 3 June 2002
Randomized isomorphic Dvoretzky theorem
Version aléatoire isomorphique du théorème de Dvoretzky
 

Alexander Litvak a , Piotr Mankiewicz b , Nicole Tomczak-Jaegermann a
a Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1, Canada 
b Institute of Mathematics, Polish Academy of Sciences, PB 137, 00-950 Warsaw, Poland 

Note presented by Gilles Pisier

Abstract

Let K be a symmetric convex body in RN for which B 2 N is the ellipsoid of minimal volume. We provide estimates for the geometric distance of a ‘typical' rank n projection of K to B 2 n , for 1⩽n <N . Known examples show that the resulting estimates are optimal (up to numerical constants) even for the Banach-Mazur distance. To cite this article: A. Litvak et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 345-350.

The full text of this article is available in PDF format.
Résumé

Soit K un corps convexe symétrique de RN dont l'ellipsoı̈de de volume minimal le contenant est la boule euclidienne B 2 N . Nous estimons la distance géométrique de projections « typiques » de rang n de K à la boule B 2 n pour tout n {1,...,N −1} (i.e. nous prouvons qu'il en existe une grande proportion au sens de la mesure de Haar normalisée sur la grassmanienne). Des exemples bien connus permettent de dire que ces estimations sont optimales (à des constantes numériques près), même pour la distance de Banach-Mazur. Pour citer cet article : A. Litvak et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 345-350.

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