[Version aléatoire isomorphique du théorème de Dvoretzky]
Soit K un corps convexe symétrique de dont l'ellipsoı̈de de volume minimal le contenant est la boule euclidienne B2N. 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 for which B2N 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.
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Alexander Litvak 1 ; Piotr Mankiewicz 2 ; Nicole Tomczak-Jaegermann 1
@article{CRMATH_2002__335_4_345_0,
author = {Alexander Litvak and Piotr Mankiewicz and Nicole Tomczak-Jaegermann},
title = {Randomized isomorphic {Dvoretzky} theorem},
journal = {Comptes Rendus. Math\'ematique},
pages = {345--350},
publisher = {Elsevier},
volume = {335},
number = {4},
year = {2002},
doi = {10.1016/S1631-073X(02)02476-7},
language = {en},
}
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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