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

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.

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.

@article{AFST_2006_6_15_4_613_0,
     author = {Cordero-Erausquin, Dario and McCann, Robert J. and Schmuckenschl\"ager, Michael},
     title = {Pr\'ekopa--Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 15},
     number = {4},
     year = {2006},
     pages = {613-635},
     doi = {10.5802/afst.1132},
     mrnumber = {2295207},
     zbl = {1125.58007},
     language = {en},
     url = {http://www.numdam.org/item/AFST_2006_6_15_4_613_0}
}
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, Serie 6, Volume 15 (2006) no. 4, pp. 613-635. doi : 10.5802/afst.1132. http://www.numdam.org/item/AFST_2006_6_15_4_613_0/

[1] Alesker, S.; Dar, S.; Milman, V. A remarkable measure preserving diffeomorphism between two convex bodies in n ,, Geom. Dedicata, Tome 74 (1999), pp. 201-212 | MR 1674116 | Zbl 0927.52007

[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 1162.35349

[3] Bakry, D.; Emery, M. Séminaire de Probabilités, Diffusions hypercontractives, Springer (Lecture Notes in Math) Tome 1123 (1985), pp. 177-206 | Numdam | MR 889476 | Zbl 0561.60080

[4] Ball, K.M. An elementary introduction to modern convex geometry, Flavors of geometry, Math. Sci. Res. Inst. Publ. (1997), pp. 1-58 | MR 1491097 | Zbl 0901.52002

[5] Barthe, F. On a reverse form of the Brascamp-Lieb inequality, Invent. Math., Tome 134 (1998) no. 2, pp. 335-361 | MR 1650312 | Zbl 0901.26010

[6] Bobkov, S.; Ledoux, M. From Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities, Geom. Funct. Anal., Tome 10 (2000), pp. 1028-1052 | MR 1800062 | Zbl 0969.26019

[7] Bobkov, S.; Gentil, I.; Ledoux, M. Hypercontractivity of Hamilton-Jacobi equations, J. Math. Pures Appl., Tome 80 (2001) no. 7, pp. 669-696 | MR 1846020 | Zbl 1038.35020

[8] Borell, C. Convex set functions in d-space, Period. Math. Hungar., Tome 6 (1975), pp. 111-136 | MR 404559 | Zbl 0274.28009

[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., Tome 22 (1976), pp. 366-389 | MR 450480 | Zbl 0334.26009

[10] Brenier, Y. Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math., Tome 44 (1991), pp. 375-417 | MR 1100809 | Zbl 0738.46011

[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 2209130 | Zbl 1082.76105

[12] Chavel, I. Riemannian Geometry—a Modern Introduction, Cambridge Tracts in Math, Tome 108 (1993) | MR 1271141 | Zbl 0810.53001

[13] Cordero-Erausquin, D. Some applications of mass transport to Gaussian type inequalities, Arch. Rational Mech. Anal., Tome 161 (2002) no. 257–269 | MR 1894593 | Zbl 0998.60080

[14] Cordero-Erausquin, D.; Mccann, R.J.; Schmuckenschläger, M. A Riemannian interpolation inequality à la Borell, Brascamp and Lieb, Invent. Math., Tome 146 (2001), pp. 219-257 | MR 1865396 | Zbl 1026.58018

[15] Cordero-Erausquin, D.; Nazaret, B.; Villani, C. A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities, Adv. Math., Tome 182 (2004) no. 2, pp. 307-332 | MR 2032031 | Zbl 1048.26010

[16] Gupta, S. Das Brunn-Minkowski inequality and its aftermath, J. Multivariate Anal. (1980) | MR 588074 | Zbl 0467.26008

[17] Gallot, S.; Hulin, D.; Lafontaine, J. Riemannian Geometry, Springer-Verlag (1990) | MR 1083149 | Zbl 0716.53001

[18] Gardner, R.J. The Brunn-Minkowski inequality, Bull. Amer. Math. Soc., Tome 39 (2002) no. 3, pp. 355-405 | MR 1898210 | Zbl 1019.26008

[19] Gromov, M.; Milman, V. A topological application of the isoperimetric inequality, Amer. J. Math., Tome 105 (1983), pp. 843-854 | MR 708367 | Zbl 0522.53039

[20] Knothe, H. Contributions to the theory of convex bodies, Michigan Math. J., Tome 4 (1957), pp. 39-52 | MR 83759 | Zbl 0077.35803

[21] Ledoux, M. Concentration of measure and logarithmic Sobolev inequalities, Séminaire de Probabilités, Tome 33 (1999), pp. 120-216 | Numdam | MR 1767995 | Zbl 0957.60016

[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 1849347 | Zbl 0995.60002

[24] Leindler, L. On a certain converse of Hölder’s inequality, Acta Sci. Math., Tome 33 (1972), pp. 217-233 | MR 2199372 | Zbl 0245.26011

[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 02183024

[27] Maurey, B. Some deviation inequalities, Geom. Funct. Anal., Tome 1 (1991), pp. 188-197 | MR 1097258 | Zbl 0756.60018

[28] Maurey, B. Inégalité de Brunn-Minkowski-Lusternik, et autres inégalités géométriques et fonctionnelles, Séminaire Bourbaki (2003) | Numdam | MR 2167203 | Zbl 02213915

[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., Tome 80 (1995), pp. 309-323 | MR 1369395 | Zbl 0873.28009

[31] Mccann, R.J. A convexity principle for interacting gases, Adv. Math., Tome 128 (1997), pp. 153-179 | MR 1451422 | Zbl 0901.49012

[32] Mccann, R.J. Polar factorization of maps on Riemannian manifolds, Geom. Funct. Anal., Tome 11 (2001) no. 3, pp. 589-608 | MR 1844080 | Zbl 1011.58009

[33] Milman, V.D.; Schechtman, G. Asymptotic theory of finite-dimensional normed spaces, Springer-Verlag, Berlin (1986) | MR 856576

[34] Otto, F. The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations, Tome 26 (2001) no. 1-2, pp. 101-174 | MR 1842429 | Zbl 0984.35089

[35] Otto, F.; Villani, C. Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal., Tome 173 (2000), pp. 361-400 | MR 1760620 | Zbl 0985.58019

[36] Prékopa, A. Logarithmic concave measures with application to stochastic programming, Acta Sci. Math., Tome 32 (1971), pp. 301-315 | MR 315079 | Zbl 0235.90044

[37] Prékopa, A. On logarithmic concave measures and functions, Acta Sci. Math. (Szeged), Tome 34 (1973), pp. 335-343 | MR 404557 | Zbl 0264.90038

[38] Schmuckenschläger, M. A concentration of measure phenomenon on uniformly convex bodies, GAFA Seminar (1992-1994), Birkaäuser (1995), pp. 275-287 | MR 1353466 | Zbl 0828.52004

[39] Schneider, R. Convex Bodies: the Brunn-Minkowski Theory, Cambridge University Press, Cambridge (1993) | MR 1216521 | Zbl 0798.52001

[40] Sturm, K.-T. Convex functionals of probability measures and nonlinear diffusions, J. Math. Pures Appl., Tome 84 (2005) | MR 2118836 | Zbl 02164960

[41] Sturm, K.-T.; Von Renesse, M.-K. Transport inequalities, gradient estimates, entropy and Ricci curvature, Comm. Pure Appl. Math., Tome 58 (2005), pp. 923-940 | MR 2142879 | Zbl 1078.53028

[42] Trudinger, N.S. Isoperimetric inequalities for quermassintegrals, Ann. Inst. H. Poincaré Anal. Non Linéaire, Tome 11 (1994), pp. 411-425 | Numdam | MR 1287239 | Zbl 0859.52001

[43] Villani, C. Graduate Studies in Math., Topics in Optimal Transportation, American Mathematical Society, Providence, RI, Tome 58 (2003) | MR 1964483 | Zbl 01909499