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.

Résumé
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”.

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”

EuDML MR Zbl | 6 citations dans Numdam

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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Maurey, Bernard. Inégalité de Brunn-Minkowski-Lusternik, et autres inégalités géométriques et fonctionnelles, dans Séminaire Bourbaki : volume 2003/2004, exposés 924-937, Astérisque, no. 299 (2005), Exposé no. 928, pp. 95-113. http://www.numdam.org/item/SB_2003-2004__46__95_0/

Bibliographie
Cité par

[And] T. W. Anderson. The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities. Proc. Amer. Math. Soc., 6 :170-176, 1955. | MR | Zbl

[ABBN] S. Artstein, K. Ball, F. Barthe, and A. Naor. Solution of Shannon's problem on the monotonicity of entropy. J. Amer. Math. Soc., 17 :975-982, 2004. | MR | Zbl

[Ba1] K. Ball. Cube slicing in $ℝ^{n}$ . Proc. Amer. Math. Soc., 97 :465-473, 1986. | MR | Zbl

[Ba2] K. Ball. Volumes of sections of cubes and related problems. In Geometric aspects of functional analysis (1987-88), volume 1376 of Lect. Notes in Math., pages 251-260. Springer. | MR | Zbl

[Ba3] K. Ball. Volume ratios and a reverse isoperimetric inequality. J. London Math. Soc. (2), 44 :351-359, 1991. | MR | Zbl

[Ba4] K. Ball. Convex geometry and functional analysis. In Handbook of the Geometry of Banach spaces, volume 1, pages 161-194. North Holland, 2001. | MR | Zbl

[BBN] K. Ball, F. Barthe, and A. Naor. Entropy jumps in the presence of a spectral gap. Duke Math. J., 119 :41-63, 2003. | MR | Zbl

[Bar] F. Barthe. On a reverse form of the Brascamp-Lieb inequality. Invent. Math., 134 :335-361, 1998. | MR | Zbl

[BaC] F. Barthe and D. Cordero-Erausquin. Inverse Brascamp-Lieb inequalities along the Heat equation. In Geometric Aspects of Functional Analysis, Israel Seminar 2002-2003, volume 1850 of Lect. Notes in Math., pages 65-71. Springer, 2004. | MR | Zbl

[Ber] B. Berndtsson. Prékopa's theorem and Kiselman's minimum principle for plurisubharmonic functions. Math. Ann., 312 :785-792, 1998. | MR | Zbl

[Bob] S. G. Bobkov. An isoperimetric inequality on the discrete cube, and an elementary proof of the isoperimetric inequality in Gauss space. Ann. Probab., 25 :206-214, 1997. | MR | Zbl

[BoL] S. G. Bobkov and M. Ledoux. From Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities. Geom. Funct. Anal., 10 :1028-1052, 2000. | MR | Zbl

[Bo1] C. Borell. The Brunn-Minkowski inequality in Gauss space. Invent. Math., 30 :207-216, 1975. | MR | Zbl

[Bo2] C. Borell. The Ehrhard inequality. C. R. Acad. Sci. Paris Sér. I Math., 337 :663-666, 2003. | MR | Zbl

[BrL] H. J. Brascamp and E. H. Lieb. Best constants in Young's inequality, its converse, and its generalization to more than three functions. Adv. in Math., 20 :151-173, 1976. | MR | Zbl

[Bre] Y. Brenier. Polar factorization and monotone rearrangement of vector-valued functions. Comm. Pure Appl. Math., 44 :375-417, 1991. | MR | Zbl

[Ca1] L. A. Caffarelli. The regularity of mappings with a convex potential. J. Amer. Math. Soc., 5 :99-104, 1992. | MR | Zbl

[Ca2] L. A. Caffarelli. Monotonicity properties of optimal transportation and the FKG and related inequalities. Comm. Math. Phys., 214 :547-563, 2000. Erratum, ibid. 225 (2002), p. 449-450. | MR | Zbl

[CHL] M. Capitaine, E. P. Hsu, and M. Ledoux. Martingale representation and a simple proof of logarithmic Sobolev inequalities on path spaces. Electron. Comm. Probab., 2 :71-81, 1997. | MR | Zbl

[Co1] D. Cordero-Erausquin. Some applications of mass transport to Gaussian-type inequalities. Arch. Rational Mech. Anal., 161 :257-269, 2002. | MR | Zbl

[Co2] D. Cordero-Erausquin. Santaló’s inequality on $ℂ^{n}$ by complex interpolation. C. R. Acad. Sci. Paris Sér. I Math., 334 :767-772, 2002. | MR | Zbl

[CFM] D. Cordero-Erausquin, M. Fradelizi, and B. Maurey. The (B) conjecture for the Gaussian measure of dilates of symmetric convex sets and related problems. J. Funct. Anal., 214 :410-427, 2004. | MR | Zbl

[CMS] D. Cordero-Erausquin, R. Mccann, and M. Schmuckenschläger. A Riemannian interpolation inequality à la Borell, Brascamp and Lieb. Invent. Math., 146 :219-257, 2001. | MR | Zbl

[CNV] D. Cordero-Erausquin, B. Nazaret, and C. Villani. A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities. Adv. in Math., 182 :307-332, 2004. | MR | Zbl

[DaG] S. Das Gupta. Brunn-Minkowski inequality and its aftermath. J. Multivariate Anal., 10 :296-318, 1980. | MR | Zbl

[Ehr] A. Ehrhard. Symétrisation dans l'espace de Gauss. Math. Scand., 53 :281-301, 1983. | MR | Zbl

[Gar] R. J. Gardner. The Brunn-Minkowski inequality. Bull. Amer. Math. Soc. (N.S.), 39 :355-405, 2002. | MR | Zbl

[HaO] H. Hadwiger and D. Ohmann. Brunn-Minkowskischer Satz und Isoperimetrie. Math. Z., 66 :1-8, 1956. | MR | Zbl

[Ha1] G. Hargé. A particular case of correlation inequality for the Gaussian measure. Ann. Probab., 27 :1939-1951, 1999. | MR | Zbl

[Ha2] G. Hargé. Inequalities for the Gaussian measure and an application to Wiener space. C. R. Acad. Sci. Paris Sér. I Math., 333 :791-794, 2001. | MR | Zbl

[HeM] R. Henstock and A. M. Macbeath. On the measure of sum-sets. I. The theorems of Brunn, Minkowski, and Lusternik. Proc. London Math. Soc. (3), 3 :182-194, 1953. | MR | Zbl

[Kan] M. Kanter. Unimodality and dominance for symmetric random vectors. Trans. Amer. Math. Soc., 229 :65-85, 1977. | MR | Zbl

[Kno] H. Knothe. Contributions to the theory of convex bodies. Michigan Math. J., 4 :39-52, 1957. | MR | Zbl

[Lat] R. Latała. A note on the Ehrhard inequality. Studia Math., 118 :169-174, 1996. | MR | Zbl

[LaO] R. Latała and K. Oleszkiewicz. Gaussian measures of dilations of convex symmetric sets. Ann. Probab., 27 :1922-1938, 1999. | MR | Zbl

[Led] M. Ledoux. The concentration of measure phenomenon. American Mathematical Society, 2001. | MR | Zbl

[Lei] L. Leindler. On a certain converse of Hölder's inequality. Acta Sci. Math. (Szeged), 33 :217-223, 1972. | MR | Zbl

[Lie] E. H. Lieb. Gaussian kernels have only Gaussian maximizers. Invent. Math., 102 :179-208, 1990. | MR | Zbl

[Lus] L. Lusternik. Die Brunn-Minkowskische Ungleichung für beliebige messbare Mengen. Dokl. Akad. Nauk SSSR, (3) :55-58, 1935. | JFM | Zbl

[MC1] R. J. Mccann. Existence and uniqueness of monotone measure-preserving maps. Duke Math. J., 80 :309-323, 1995. | MR | Zbl

[MC2] R. J. Mccann. Polar factorization of maps on Riemannian manifolds. Geom. Funct. Anal., 11 :589-608, 2001. | MR | Zbl

[MeP] M. Meyer and A. Pajor. Sections of the unit ball of $ℓ_{n}^{p}$ . J. Funct. Anal., 80 :109-123, 1988. | MR | Zbl

[Mon] G. Monge. Mémoire sur la théorie des déblais et des remblais. In Histoire de l'Académie Royale des Sciences de Paris, avec les Mémoires de Mathématique et de Physique pour la même année, pages 666-704. 1781.

[Pis] G. Pisier. The volume of convex bodies and Banach space geometry, volume 94 of Cambridge Tracts in Mathematics. Cambridge University Press, 1989. | MR | Zbl

[Pit] L. D. Pitt. A Gaussian correlation inequality for symmetric convex sets. Ann. Probab., 5 :470-474, 1977. | MR | Zbl

[Pre] A. Prékopa. On logarithmic concave measures and functions. Acta Sci. Math. (Szeged), 34 :335-343, 1973. | MR | Zbl

[Sim] C. G. Simader. Essential self-adjointness of Schrödinger operators bounded from below. Math. Z., 159 :47-50, 1978. | MR | Zbl

[SuT] V. N. Sudakov and B. S. Tsirel'Son. Extremal properties of half-spaces for spherically invariant measures. J. Soviet Math., 9 :9-18, 1978. traduit de Zap. Nauch. Sem. L.O.M.I. 41 (1974), p. 14-24. | MR | Zbl

[Vil] C. Villani. Topics in Optimal Transportation, volume 58 of Graduate Studies in Mathematics. American Mathematical Society, 2003. | MR | Zbl