Quelques exemples d'application du transport de mesure en géométrie euclidienne et riemannienne
Séminaire de théorie spectrale et géométrie, Tome 22 (2003-2004), pp. 125-152. 
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     title = {Quelques exemples d'application du transport de mesure en g\'eom\'etrie euclidienne et riemannienne},
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Cordero-Erausquin, Dario. Quelques exemples d'application du transport de mesure en géométrie euclidienne et riemannienne. Séminaire de théorie spectrale et géométrie, Tome 22 (2003-2004), pp. 125-152. http://archive.numdam.org/item/TSG_2003-2004__22__125_0/

[1] M. Agueh, N. Ghoussoub ET X. Kang, Geometric inequalities via a general comparison principlefor interacting gases, Geom. Funct. Anal., 14 ( 2004), 215-244. | MR | Zbl

[2] A. Alvino, V. Ferone, G. Trombetti, ET P.-L. Lions, Convex symmetrization and applications. Ann. Inst. H. Poincaré Anal Non Linéaire, 14, ( 1997), 275-293. | Numdam | MR | Zbl

[3] T. Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geometry 11,4 ( 1976), 573-598. | MR | Zbl

[4] D. Bakry, ET M. Émery Diffusions hypercontractives, in Séminaire de Probabilités XIX, Lecture Notes in Math., 1123, Springer ( 1985), 177-206. | Numdam | MR | Zbl

[5] K.M. Ball, An elementary introduction to modem convex geometry, in Flavors of geometry, Math. Sci. Res. Inst. Publ. 31, Cambridge Univ. Press ( 1997),1-58. | Zbl

[6] F. Barthe, On a reverse form of the Brascamp-Lieb inequality, Invent. Math., 134 ( 1998), n°2, 335-361. | MR | Zbl

[7] F. Barthe, Optimal Young's inequality and its converse : a simple proof, Geom. Funct. Anal, 8 ( 1998), 234-242. | MR | Zbl

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

[9] S. Bobkov, I. Gentil ET M. Ledoux, Hypercontractivity of Hamilton-Jacobi equations, J. Math. Pures Appl (9) 80 ( 2001), no4, 669-696. | MR | Zbl

[10] C. Borell, Convex set functions in d-space, Period. Math. Hungar.,6 ( 1975), 111-136. | MR | Zbl

[11] Hj. Brascamp ET E.H. Lieb, On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Funct. Anal, 22 ( 1976), 366-389. | MR | Zbl

[12] Y. Brenier, Décomposition polaire et réarrangement monotone des champs de vecteurs, C.R. Acad. Sci. Paris Sér. I Math., 305 ( 1987), 805-808. | MR | Zbl

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

[14] J. Brothers ET W. Ziemer Minimal rearrangements of Sobolev functions, J. Reine Angew. Math., 384 ( 1988), 153-179. | MR | Zbl

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

[16] D. Cordero -Erausquin, W. Gangbo, C. Houdré, Inequalities for generalized entropy and optimal transportation, in Recent Advances in the Theory and Applications of Mass Transport, Contemp. Math., 353, A.M.S., 2004. | MR | Zbl

[17] D. Cordero -Erausquin, Rj. Mccann ET M. Schmuckenschläger, A Riemannian interpolation inequality à la Borell, Brascamp and Lieb, Invent Math., 146 ( 2001), 219-257. | MR | Zbl

[18] D. Cordero -Erausquin, Rj. Mccann ET M. Schmuckenschläger, Prékopa-Leindler type inequalities on Riemannian manifolds, mass transport and Jacobi fields, Preprint ( 2004). | MR

[19] D. Cordero -Erausquin, B. Nazaret ET C. Villani, A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities, Adv. Math., 182 ( 2004), n°2, 307-332. | MR | Zbl

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

[21] O. Druet ET E. Hebey, The AB program in geometric analysis : sharp Sobolev inequalities and related problems, Mem. Amer.Math. Soc., 160 ( 2002), n°761. | MR | Zbl

[22] S. Gallot, D. Hulin ET J. Lafontaine, Riemannian Geometry, Springer-Verlag (Universitext), Berlin, 1990. | MR | Zbl

[23] Rj. Gardner, The Brunn-Minkowski inequality, Bull. Amer. Math. Soc. (N.S.), 39-3 ( 2002), 355-405. | MR | Zbl

[24] M. Gromov ET V. Milman, A topological application of the isoperimetric inequatliry, Amer. J. Math., 105 ( 1983), 843-854. | MR | Zbl

[25] H. Hadwiger ET D. Ohmann, Brunn-Minkowskischer Satz und Isoperimetrie, Math. Z, 66 ( 1956), 1-8. | MR | Zbl

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

[27] M. Ledoux, Concentration of measure and logarithmic Sobolev inequalities, in Séminaire de Probabilités XXXIII, Lecture Notes in Math. 1709, Springer ( 1999), 120-216. | Numdam | MR | Zbl

[28] M. Ledoux, The geometry of Markov diffusion generators, Ann. Fac. Sci. Toulouse Math. (6)9 ( 2000), n°2, 305-366. | Numdam | MR | Zbl

[29] M. Ledoux, Measure concentration, transportation cost, and functional inequalities, Summer School on Singular Phenomena and Scaling in Mathematica! Models, Bonn, 10-13 June 2003 (http ://www.lsp.ups-tlse.fr/Ledoux).

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

[31] L. Leindler, On a certain converse of Hölder's inequality, Acta Sci. Math., 33 ( 1972), 217-233. | MR | Zbl

[32] F. Maggi ET C. Villani, Balls have the worst best Sobolev inequality, Preprint ( 2004). | Zbl

[33] B. Maurey, Some deviation inequalities, Geom.Funct. Anal., 1 ( 1991), 188-197. | MR | Zbl

[34] B. Maurey, Inégalité de Brunn-Minkowski-Lusternik, et autres inégalités géométriques et fonctionnelles, Séminaire Bourbaki, Novembre 2003. | Numdam | MR | Zbl

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

[36] R.J. Mccann, A convexity principle for interacting gases, Adv. Math., 128 ( 1997), 153-179. | MR | Zbl

[37] R.J. Mccann, Polar factorizatîon of maps on Riemannian manifolds, Geom. Funct. Anal., 11 ( 2001), n°3, 589-608. | MR | Zbl

[38] V. Milman ET G. Schechtman, Asymptotic theory of finite-dimensional normed spaces, LNM n°1200, Springer, Berlin, 1986. With an appendix by M. Gromov. | MR | Zbl

[39] G. Monge, Mémoire sur la théorie des déblais et des remblais, in Histoire de l'Académie Royale des Sciences (Année 1781), Imprimerie Royale, Paris ( 1784), 666-704.

[40] F. Otto, The geometry of dissipative evolution equations : the porous medium equation, Comm. Partial Differential Equations, 26 ( 2001), n° 1-2,101-174. | MR | Zbl

[41] F. Otto ET C. Villani, Generalization of an inequality byTalagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal., 173 ( 2000), 361-400. | MR | Zbl

[42] A. Prékopa, Logarithmic concave measures with application to stochastic programming, Acta Sci. Math., 32 ( 1971), 301-315. | MR | Zbl

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

[44] R. Schneider, Convex Bodies : the Brunn-Minkowski Theory, Cambridge University Press, Cambridge, 1993. | MR | Zbl

[45] K.-T. Sturm ET M.-K. Von Renesse, Transport inequalities, gradient estimates, entropy and Ricci curvature, à paraître dans Comm. Pure Appl. Math. | MR | Zbl

[46] G. Talenti, Best constants in Sobolev inequality, Ann. Mat Pura Appl. (IV), 110 ( 1976), 353-372. | MR | Zbl

[47] C. Villani, Topicsin Optimal Transportation, Graduate Studies in Math. 58, American Mathematical Society, Providence, RI, 2003. | MR | Zbl