Prékopa–Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 15 (2006) no. 4, pp. 613-635.

Nous étudions l’extension d’inégalités de type Prékopa- Leindler au cas d’une variété riemannienne M équipée d’une mesure ayant une densité e -V où le potentiel V et la courbure de Ricci vérifient Hess x V+Ric x λI(xM), pour un certain λ. Nous ferons appel, comme dans notre travail précédent [14], au transport optimal de mesure. Mais nous exploiterons plus encore son lien avec les champs de Jacobi, ce qui permettra de ramener la discussion à l’étude du déterminant d’une matrice de champs de Jacobi. Nous présentons également d’autres applications de la méthode, en particulier aux inégalités de Sobolev logarithmiques (critère de Bakry-Emery) et à l’étude de la convexité de déplacement de la fonctionnelle entropie.

We investigate Prékopa-Leindler type inequalities on a Riemannian manifold M equipped with a measure with density e -V where the potential V and the Ricci curvature satisfy Hess x V+Ric x λI for all xM, with some λ. As in our earlier work [14], the argument uses optimal mass transport on M, but here, with a special emphasis on its connection with Jacobi fields. A key role will be played by the differential equation satisfied by the determinant of a matrix of Jacobi fields. We also present applications of the method to logarithmic Sobolev inequalities (the Bakry-Emery criterion will be recovered) and to transport inequalities. A study of the displacement convexity of the entropy functional completes the exposition.

DOI : 10.5802/afst.1132
Cordero-Erausquin, Dario 1 ; McCann, Robert J. 2 ; Schmuckenschläger, Michael 3

1 Laboratoire d’Analyse et de Mathématiques Appliquées, Université de Marne la Vallée, 77454 Marne la Vallée Cedex 2, France.
2 Department of Mathematics, University of Toronto, Toronto Ontario Canada M5S 3G3.
3 Institut für Analysis und Numerik, Universität Linz, A-4040 Linz, Österreich.
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     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
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Cordero-Erausquin, Dario; McCann, Robert J.; Schmuckenschläger, Michael. Prékopa–Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 15 (2006) no. 4, pp. 613-635. doi : 10.5802/afst.1132. http://archive.numdam.org/articles/10.5802/afst.1132/

[1] Alesker, S.; Dar, S.; Milman, V. A remarkable measure preserving diffeomorphism between two convex bodies in n ,, Geom. Dedicata, Volume 74 (1999), pp. 201-212 | MR | Zbl

[2] Ambrosio, L.A.; Gigli, N.; Savaré, G. Gradient flows with metric and differentiable structures,and applications to the Wasserstein space (To appear in the Academy of Lincei proceedings on “Nonlinear evolution equations”, Rome) | Zbl

[3] Bakry, D.; Emery, M. Séminaire de Probabilités, Diffusions hypercontractives (Lecture Notes in Math), Volume 1123, Springer (1985), pp. 177-206 | Numdam | MR | Zbl

[4] Ball, K.M. An elementary introduction to modern convex geometry, Flavors of geometry, Math. Sci. Res. Inst. Publ. (1997), pp. 1-58 | MR | Zbl

[5] Barthe, F. On a reverse form of the Brascamp-Lieb inequality, Invent. Math., Volume 134 (1998) no. 2, pp. 335-361 | MR | Zbl

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

[7] Bobkov, S.; Gentil, I.; Ledoux, M. Hypercontractivity of Hamilton-Jacobi equations, J. Math. Pures Appl., Volume 80 (2001) no. 7, pp. 669-696 | MR | Zbl

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

[9] Brascamp, H.J.; Lieb, E.H. 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., Volume 22 (1976), pp. 366-389 | MR | Zbl

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

[11] Carrillo, J.A.; McCann, R.J.; Villani, C. Contractions in the 2-Wasserstein length space and thermalization of granular media (to appear in Arch. Rational Mech. Anal.) | MR | Zbl

[12] Chavel, I. Riemannian Geometry—a Modern Introduction, Cambridge Tracts in Math, Volume 108 (1993) | MR | Zbl

[13] Cordero-Erausquin, D. Some applications of mass transport to Gaussian type inequalities, Arch. Rational Mech. Anal., Volume 161 (2002) no. 257–269 | MR | Zbl

[14] Cordero-Erausquin, D.; McCann, R.J.; Schmuckenschläger, M. A Riemannian interpolation inequality à la Borell, Brascamp and Lieb, Invent. Math., Volume 146 (2001), pp. 219-257 | MR | Zbl

[15] Cordero-Erausquin, D.; Nazaret, B.; Villani, C. A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities, Adv. Math., Volume 182 (2004) no. 2, pp. 307-332 | MR | Zbl

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

[17] Gallot, S.; Hulin, D.; Lafontaine, J. Riemannian Geometry, Springer-Verlag, 1990 | MR | Zbl

[18] Gardner, R.J. The Brunn-Minkowski inequality, Bull. Amer. Math. Soc., Volume 39 (2002) no. 3, pp. 355-405 | MR | Zbl

[19] Gromov, M.; Milman, V. A topological application of the isoperimetric inequality, Amer. J. Math., Volume 105 (1983), pp. 843-854 | MR | Zbl

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

[21] Ledoux, M. Concentration of measure and logarithmic Sobolev inequalities, Séminaire de Probabilités, Volume 33 (1999), pp. 120-216 | Numdam | MR | Zbl

[22] Ledoux, M. Measure concentration, transportation cost, and functional inequalities, Summer School on Singular Phenomena and Scaling in Mathematical Models, Bonn (2003)

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

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

[25] Lott, J.; Villani, C. Ricci curvature for metric-measure spaces via optimal transport (preprint)

[26] Maggi, F.; Villani, C. Balls have the worst best Sobolev inequality (preprint) | Zbl

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

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

[29] McCann, R.J. A Convexity Principle for Interacting Gases and Equilibrium Crystals, Princeton University (1994) (Ph. D. Thesis)

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

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

[32] McCann, R.J. Polar factorization of maps on Riemannian manifolds, Geom. Funct. Anal., Volume 11 (2001) no. 3, pp. 589-608 | MR | Zbl

[33] Milman, V.D.; Schechtman, G. Asymptotic theory of finite-dimensional normed spaces, Springer-Verlag, Berlin, 1986 | MR

[34] Otto, F. The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations, Volume 26 (2001) no. 1-2, pp. 101-174 | MR | Zbl

[35] Otto, F.; Villani, C. Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal., Volume 173 (2000), pp. 361-400 | MR | Zbl

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

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

[38] Schmuckenschläger, M. A concentration of measure phenomenon on uniformly convex bodies, GAFA Seminar (1992-1994), Birkaäuser (1995), pp. 275-287 | MR | Zbl

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

[40] Sturm, K.-T. Convex functionals of probability measures and nonlinear diffusions, J. Math. Pures Appl., Volume 84 (2005) | MR | Zbl

[41] Sturm, K.-T.; von Renesse, M.-K. Transport inequalities, gradient estimates, entropy and Ricci curvature, Comm. Pure Appl. Math., Volume 58 (2005), pp. 923-940 | MR | Zbl

[42] Trudinger, N.S. Isoperimetric inequalities for quermassintegrals, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 11 (1994), pp. 411-425 | Numdam | MR | Zbl

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

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