[Réduction du problème des sections de corps convexes au cas de rapport volumique borné]
Cette Note concerne le problème bien connu de la minoration uniforme de la mesure des sections de codimension 1 de corps convexes isotrope dans , 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.
Accepté le :
Publié le :
@article{CRMATH_2003__336_4_331_0,
author = {Bourgain, Jean and Klartag, Bo'az and Milman, Vitali},
title = {A reduction of the slicing problem to finite volume ratio bodies},
journal = {Comptes Rendus. Math\'ematique},
pages = {331--334},
publisher = {Elsevier},
volume = {336},
number = {4},
year = {2003},
doi = {10.1016/S1631-073X(03)00041-4},
language = {en},
url = {http://archive.numdam.org/articles/10.1016/S1631-073X(03)00041-4/}
}
TY - JOUR AU - Bourgain, Jean AU - Klartag, Bo'az AU - Milman, Vitali TI - A reduction of the slicing problem to finite volume ratio bodies JO - Comptes Rendus. Mathématique PY - 2003 SP - 331 EP - 334 VL - 336 IS - 4 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/S1631-073X(03)00041-4/ DO - 10.1016/S1631-073X(03)00041-4 LA - en ID - CRMATH_2003__336_4_331_0 ER -
%0 Journal Article %A Bourgain, Jean %A Klartag, Bo'az %A Milman, Vitali %T A reduction of the slicing problem to finite volume ratio bodies %J Comptes Rendus. Mathématique %D 2003 %P 331-334 %V 336 %N 4 %I Elsevier %U http://archive.numdam.org/articles/10.1016/S1631-073X(03)00041-4/ %R 10.1016/S1631-073X(03)00041-4 %G en %F CRMATH_2003__336_4_331_0
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://archive.numdam.org/articles/10.1016/S1631-073X(03)00041-4/
[1] Normed spaces with a weak-Gordon–Lewis property, Proc. of Funct. Anal., University of Texas and Austin (1987–1989), Lecture Notes in Math., 1470, Springer, 1991, pp. 36-47
[2] On the distribution of polynomials on high dimensional convex sets, Geometric aspects of functional analysis (1989–90), Lecture Notes in Math., 1469, Springer, 1991, pp. 127-137
[3] Remarks on Bourgain's problem on slicing of convex bodies, Geometric aspects of functional analysis, Oper. Theory Adv. Appl., 77, Birkhäuser, 1995, pp. 61-66
[4] Inégalité de Brunn–Minkowski inverse et applications à la théorie locale des espaces normés, C. R. Acad. Sci. Paris, Sér. 1, Volume 320 (1986), pp. 25-28
[5] Geometrical inequalities and mixed volumes in the local theory of Banach spaces, Colloque en l'honneur de Laurent Schwartz, Astérisque, Volume 131 (1985) no. 1, pp. 373-400
[6] Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space, Geometric aspects of functional analysis (1987–88), Lecture Notes in Math., 1376, Springer, 1989, pp. 64-104
[7] The Volume of Convex Bodies and Banach Space Geometry, Cambridge Tracts in Math., 94, Cambridge University Press, 1997
Cité par Sources :
