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.
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.
Accepted:
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Jean Bourgain 1; Bo'az Klartag 2; Vitali Milman 2
@article{CRMATH_2003__336_4_331_0, author = {Jean Bourgain and Bo'az Klartag and Vitali Milman}, 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}, }
TY - JOUR AU - Jean Bourgain AU - Bo'az Klartag AU - Vitali Milman 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 DO - 10.1016/S1631-073X(03)00041-4 LA - en ID - CRMATH_2003__336_4_331_0 ER -
Jean Bourgain; Bo'az Klartag; Vitali Milman. A reduction of the slicing problem to finite volume ratio bodies. Comptes Rendus. Mathématique, Volume 336 (2003) no. 4, pp. 331-334. doi : 10.1016/S1631-073X(03)00041-4. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00041-4/
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