Inégalité de Brunn-Minkowski-Lusternik, et autres inégalités géométriques et fonctionnelles

Maurey, Bernard

Séminaire Bourbaki : volume 2003/2004, exposés 924-937, Astérisque, no. 299 (2005), Exposé no. 928, pp. 95-113.

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”.

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”

Classification : 26D15, 39B62, 52A40, 46Bxx, 60E15, 60G15
Mot clés : inégalité de Brunn-Minkowski, inégalité de Prékopa-Leindler, inégalité de Brascamp-Lieb, inégalité isopérimétrique, inégalité de Sobolev, fonction log-concave, mesure log-concave, corps convexe, transport de mesure, application de Brenier, mesure gaussienne, inégalité de déviation, interpolation complexe
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

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