Geometry/Functional Analysis
A reduction of the slicing problem to finite volume ratio bodies
[Réduction du problème des sections de corps convexes au cas de rapport volumique borné]
Comptes Rendus. Mathématique, Tome 336 (2003) no. 4, pp. 331-334.

Cette Note concerne le problème bien connu de la minoration uniforme de la mesure des sections de codimension 1 de corps convexes isotrope dans n , ce qui équivaut à une borne uniforme de la constante d'isotropie. Nous démontrons qu'une réponse affirmative à cette question dans le cas particulier d'un corps à rapport volumique borné (c'est-à-dire tel que la racine n-ième du volume de l'ellipsoide de John admet une borne inférieure) entraı̂ne une réponse affirmative en général. La méthode utilise des techniques de symétrisation et de géométrie des espaces de Banach.

Here we discuss results around the slicing problem, which is a well known open problem in asymptotic convex geometry. We show that if one can prove that the isotropic constant of bodies with a finite volume ratio is uniformly bounded – then it would follow that the isotropic constant of any convex body is uniformly bounded.

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Accepté le :
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DOI : 10.1016/S1631-073X(03)00041-4
Bourgain, Jean 1 ; Klartag, Bo'az 2 ; Milman, Vitali 2

1 Institute for Advanced Study, Olden Lane, Princeton, NJ 08540, USA
2 School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel
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Bourgain, Jean; Klartag, Bo'az; Milman, Vitali. A reduction of the slicing problem to finite volume ratio bodies. Comptes Rendus. Mathématique, Tome 336 (2003) no. 4, pp. 331-334. doi : 10.1016/S1631-073X(03)00041-4. http://www.numdam.org/articles/10.1016/S1631-073X(03)00041-4/

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