[Concentration et isopérimétrie sont équivalentes lorsque la courbure est bornée inférieurement]
Il est connu que les inégalités isopérimétriques impliquent dans un cadre espace-mesure-métrique très général des inégalités de concentration correspondantes. Les premières bornent la mesure de bord d'un ensemble en fonction de sa mesure, tandis que les dernières bornent la mesure d'un ensemble qui est séparé d'un ensemble ayant la moitié de la mesure totale, en fonction de leur distance mutuelle. Nous montrons que sous une condition de borne inférieure sur le tenseur de courbure de Bakry–Émery d'une variété riemannienne équipée d'une densité, les inégalités de concentration complètement générales impliquent inversement leurs analogues isopérimétriques, à des constantes près qui sont indépendantes de la dimension. Notre méthode est entièrement géométrique, continuant l'approche de Gromov, Buser et Morgan.
It is well known that isoperimetric inequalities imply in a very general measure-metric-space setting appropriate concentration inequalities. The former bound the boundary measure of sets as a function of their measure, whereas the latter bound the measure of sets separated from sets having half the total measure, as a function of their mutual distance. We show that under a lower bound condition on the Bakry–Émery curvature tensor of a Riemannian manifold equipped with a density, completely general concentration inequalities imply back their isoperimetric counterparts, up to dimension independent bounds. Our method is entirely geometric, continuing the approach of Gromov, Buser and Morgan.
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@article{CRMATH_2009__347_1-2_73_0,
author = {Milman, Emanuel},
title = {Concentration and isoperimetry are equivalent assuming curvature lower bound},
journal = {Comptes Rendus. Math\'ematique},
pages = {73--76},
publisher = {Elsevier},
volume = {347},
number = {1-2},
year = {2009},
doi = {10.1016/j.crma.2008.11.014},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2008.11.014/}
}
TY - JOUR AU - Milman, Emanuel TI - Concentration and isoperimetry are equivalent assuming curvature lower bound JO - Comptes Rendus. Mathématique PY - 2009 SP - 73 EP - 76 VL - 347 IS - 1-2 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2008.11.014/ DO - 10.1016/j.crma.2008.11.014 LA - en ID - CRMATH_2009__347_1-2_73_0 ER -
%0 Journal Article %A Milman, Emanuel %T Concentration and isoperimetry are equivalent assuming curvature lower bound %J Comptes Rendus. Mathématique %D 2009 %P 73-76 %V 347 %N 1-2 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2008.11.014/ %R 10.1016/j.crma.2008.11.014 %G en %F CRMATH_2009__347_1-2_73_0
Milman, Emanuel. Concentration and isoperimetry are equivalent assuming curvature lower bound. Comptes Rendus. Mathématique, Tome 347 (2009) no. 1-2, pp. 73-76. doi : 10.1016/j.crma.2008.11.014. http://www.numdam.org/articles/10.1016/j.crma.2008.11.014/
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