Randomized isomorphic Dvoretzky theorem

Alexander Litvak; Piotr Mankiewicz; Nicole Tomczak-Jaegermann

doi:10.1016/S1631-073X(02)02476-7

Randomized isomorphic Dvoretzky theorem
[Version aléatoire isomorphique du théorème de Dvoretzky]

Alexander Litvak ¹ ; Piotr Mankiewicz ² ; Nicole Tomczak-Jaegermann ¹

¹ Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1, Canada
² Institute of Mathematics, Polish Academy of Sciences, PB 137, 00-950 Warsaw, Poland

Comptes Rendus. Mathématique, Volume 335 (2002) no. 4, pp. 345-350.

Résumé
Abstract

Soit K un corps convexe symétrique de $ℝ^{N}$ dont l'ellipsoı̈de de volume minimal le contenant est la boule euclidienne B₂^N. Nous estimons la distance géométrique de projections « typiques » de rang n de K à la boule B₂ⁿ 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.

Let K be a symmetric convex body in $ℝ^{N}$ for which B₂^N is the ellipsoid of minimal volume. We provide estimates for the geometric distance of a ‘typical’ rank n projection of K to B₂ⁿ, for 1⩽n<N. Known examples show that the resulting estimates are optimal (up to numerical constants) even for the Banach–Mazur distance.

Reçu le : 2002-05-07
Accepté le : 2002-06-03
Publié le : 2002-01-01

DOI : 10.1016/S1631-073X(02)02476-7

Affiliations des auteurs :

Alexander Litvak ¹ ; Piotr Mankiewicz ² ; Nicole Tomczak-Jaegermann ¹

¹ Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1, Canada
² Institute of Mathematics, Polish Academy of Sciences, PB 137, 00-950 Warsaw, Poland

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     author = {Alexander Litvak and Piotr Mankiewicz and Nicole Tomczak-Jaegermann},
     title = {Randomized isomorphic {Dvoretzky} theorem},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {345--350},
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     number = {4},
     year = {2002},
     doi = {10.1016/S1631-073X(02)02476-7},
     language = {en},
}

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Alexander Litvak; Piotr Mankiewicz; Nicole Tomczak-Jaegermann. Randomized isomorphic Dvoretzky theorem. Comptes Rendus. Mathématique, Volume 335 (2002) no. 4, pp. 345-350. doi : 10.1016/S1631-073X(02)02476-7. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02476-7/

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