The Brunn-Minkowski-Lusternik inequality, and other geometric and functional inequalities

Bernard Maurey

Séminaire Bourbaki (2003-2004)

  • Volume: 46, page 95-114
  • ISSN: 0303-1179

Abstract

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The theory of convex bodies has begun by the end of the 19th century with the Brunn inequality, later generalized as Brunn-Minkowski-Lusternik inequality, that applies also to non convex sets. This subject has had for a long time contacts with isoperimetric problems and inequalities in Analysis such as Sobolev inequalities. We shall deal with some more recent aspects of geometric inequalities; some of them are related to the mass transportation technique, in particular the “Brenier map”

How to cite

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Maurey, Bernard. "Inégalité de Brunn-Minkowski-Lusternik, et autres inégalités géométriques et fonctionnelles." Séminaire Bourbaki 46 (2003-2004): 95-114. <http://eudml.org/doc/252148>.

@article{Maurey2003-2004,
abstract = {La théorie des corps convexes a commencé à la fin du xixe siècle avec l’inégalité de Brunn, généralisée ensuite sous la forme de l’inégalité de Brunn-Minkowski-Lusternik, qui s’applique à des ensembles non convexes. Ce thème a depuis longtemps des contacts avec les problèmes isopérimétriques et avec des inégalités d’Analyse telle que les plongements de Sobolev. On développera quelques aspects plus récents des inégalités géométriques, dont certains sont liés à la technique du transport de mesure, notamment le transport dit “de Brenier”.},
author = {Maurey, Bernard},
journal = {Séminaire Bourbaki},
keywords = {Brunn-Minkowski inequality; Prékopa-Leindler inequality; Brascamp-Lieb inequality; isoperimetric inequality; Sobolev inequality; log-concave function; log-concave measure; convex body; transportation of mass; Brenier map; gaussian measure; deviation inequality; complex interpolation},
language = {fre},
pages = {95-114},
publisher = {Association des amis de Nicolas Bourbaki, Société mathématique de France},
title = {Inégalité de Brunn-Minkowski-Lusternik, et autres inégalités géométriques et fonctionnelles},
url = {http://eudml.org/doc/252148},
volume = {46},
year = {2003-2004},
}

TY - JOUR
AU - Maurey, Bernard
TI - Inégalité de Brunn-Minkowski-Lusternik, et autres inégalités géométriques et fonctionnelles
JO - Séminaire Bourbaki
PY - 2003-2004
PB - Association des amis de Nicolas Bourbaki, Société mathématique de France
VL - 46
SP - 95
EP - 114
AB - La théorie des corps convexes a commencé à la fin du xixe siècle avec l’inégalité de Brunn, généralisée ensuite sous la forme de l’inégalité de Brunn-Minkowski-Lusternik, qui s’applique à des ensembles non convexes. Ce thème a depuis longtemps des contacts avec les problèmes isopérimétriques et avec des inégalités d’Analyse telle que les plongements de Sobolev. On développera quelques aspects plus récents des inégalités géométriques, dont certains sont liés à la technique du transport de mesure, notamment le transport dit “de Brenier”.
LA - fre
KW - Brunn-Minkowski inequality; Prékopa-Leindler inequality; Brascamp-Lieb inequality; isoperimetric inequality; Sobolev inequality; log-concave function; log-concave measure; convex body; transportation of mass; Brenier map; gaussian measure; deviation inequality; complex interpolation
UR - http://eudml.org/doc/252148
ER -

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Citations in EuDML Documents

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  1. Cédric Villani, Transport optimal et courbure de Ricci
  2. Cédric Villani, Transport optimal et courbure de Ricci
  3. Ivan Gentil, From the Prékopa-Leindler inequality to modified logarithmic Sobolev inequality
  4. Dario Cordero-Erausquin, Robert J. McCann, Michael Schmuckenschläger, Prékopa–Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport
  5. Dario Cordero-Erausquin, Quelques exemples d'application du transport de mesure en géométrie euclidienne et riemannienne

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