Prékopa–Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport

Dario Cordero-Erausquin[1]; Robert J. McCann[2]; Michael Schmuckenschläger[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.

Annales de la faculté des sciences de Toulouse Mathématiques (2006)

  • Volume: 15, Issue: 4, page 613-635
  • ISSN: 0240-2963

Abstract

top
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 ≥ λ I for all x ∈ M , with some λ ∈ ℝ . 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.

How to cite

top

Cordero-Erausquin, Dario, McCann, Robert J., and Schmuckenschläger, Michael. "Prékopa–Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport." Annales de la faculté des sciences de Toulouse Mathématiques 15.4 (2006): 613-635. <http://eudml.org/doc/10016>.

@article{Cordero2006,
abstract = {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 $\operatorname\{Hess\}_x V + \operatorname\{Ric\}_x \ge \lambda \, I$ for all $x\in M$, with some $\lambda \in \mathbb\{R\}$. 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.},
affiliation = {Laboratoire d’Analyse et de Mathématiques Appliquées, Université de Marne la Vallée, 77454 Marne la Vallée Cedex 2, France.; Department of Mathematics, University of Toronto, Toronto Ontario Canada M5S 3G3.; Institut für Analysis und Numerik, Universität Linz, A-4040 Linz, Österreich.},
author = {Cordero-Erausquin, Dario, McCann, Robert J., Schmuckenschläger, Michael},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
language = {eng},
number = {4},
pages = {613-635},
publisher = {Université Paul Sabatier, Toulouse},
title = {Prékopa–Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport},
url = {http://eudml.org/doc/10016},
volume = {15},
year = {2006},
}

TY - JOUR
AU - Cordero-Erausquin, Dario
AU - McCann, Robert J.
AU - Schmuckenschläger, Michael
TI - Prékopa–Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport
JO - Annales de la faculté des sciences de Toulouse Mathématiques
PY - 2006
PB - Université Paul Sabatier, Toulouse
VL - 15
IS - 4
SP - 613
EP - 635
AB - 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 $\operatorname{Hess}_x V + \operatorname{Ric}_x \ge \lambda \, I$ for all $x\in M$, with some $\lambda \in \mathbb{R}$. 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.
LA - eng
UR - http://eudml.org/doc/10016
ER -

References

top
  1. S. Alesker, S. Dar, V. Milman, A remarkable measure preserving diffeomorphism between two convex bodies in ℝ n ,, Geom. Dedicata 74 (1999), 201-212 Zbl0927.52007MR1674116
  2. L.A. Ambrosio, N. Gigli, G. Savaré, Gradient flows with metric and differentiable structures,and applications to the Wasserstein space Zbl1162.35349
  3. D. Bakry, M. Emery, Séminaire de Probabilités, Diffusions hypercontractives 1123 (1985), 177-206, Springer Zbl0561.60080MR889476
  4. K.M. Ball, An elementary introduction to modern convex geometry, Flavors of geometry, Math. Sci. Res. Inst. Publ. (1997), 1-58 Zbl0901.52002MR1491097
  5. F. Barthe, On a reverse form of the Brascamp-Lieb inequality, Invent. Math. 134 (1998), 335-361 Zbl0901.26010MR1650312
  6. S. Bobkov, M. Ledoux, From Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities, Geom. Funct. Anal. 10 (2000), 1028-1052 Zbl0969.26019MR1800062
  7. S. Bobkov, I. Gentil, M. Ledoux, Hypercontractivity of Hamilton-Jacobi equations, J. Math. Pures Appl. 80 (2001), 669-696 Zbl1038.35020MR1846020
  8. C. Borell, Convex set functions in d -space, Period. Math. Hungar. 6 (1975), 111-136 Zbl0274.28009MR404559
  9. H.J. Brascamp, 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 Zbl0334.26009MR450480
  10. Y. Brenier, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math. 44 (1991), 375-417 Zbl0738.46011MR1100809
  11. J.A. Carrillo, R.J. McCann, C. Villani, Contractions in the 2 -Wasserstein length space and thermalization of granular media Zbl1082.76105MR2209130
  12. I. Chavel, Riemannian Geometry—a Modern Introduction, Cambridge Tracts in Math 108 (1993) Zbl0810.53001MR1271141
  13. D. Cordero-Erausquin, Some applications of mass transport to Gaussian type inequalities, Arch. Rational Mech. Anal. 161 (2002) Zbl0998.60080MR1894593
  14. D. Cordero-Erausquin, R.J. McCann, M. Schmuckenschläger, A Riemannian interpolation inequality à la Borell, Brascamp and Lieb, Invent. Math. 146 (2001), 219-257 Zbl1026.58018MR1865396
  15. D. Cordero-Erausquin, B. Nazaret, C. Villani, A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities, Adv. Math. 182 (2004), 307-332 Zbl1048.26010MR2032031
  16. S. Das Gupta, Brunn-Minkowski inequality and its aftermath, J. Multivariate Anal. (1980) Zbl0467.26008MR588074
  17. S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry, (1990), Springer-Verlag Zbl0716.53001MR1083149
  18. R.J. Gardner, The Brunn-Minkowski inequality, Bull. Amer. Math. Soc. 39 (2002), 355-405 Zbl1019.26008MR1898210
  19. M. Gromov, V. Milman, A topological application of the isoperimetric inequality, Amer. J. Math. 105 (1983), 843-854 Zbl0522.53039MR708367
  20. H. Knothe, Contributions to the theory of convex bodies, Michigan Math. J. 4 (1957), 39-52 Zbl0077.35803MR83759
  21. M. Ledoux, Concentration of measure and logarithmic Sobolev inequalities, Séminaire de Probabilités 33 (1999), 120-216 Zbl0957.60016MR1767995
  22. M. Ledoux, Measure concentration, transportation cost, and functional inequalities, Summer School on Singular Phenomena and Scaling in Mathematical Models (2003), Bonn 
  23. M Ledoux, The concentration of measure phenomenon, (2001), American Mathematical Society, Providence, RI Zbl0995.60002MR1849347
  24. L. Leindler, On a certain converse of Hölder’s inequality, Acta Sci. Math. 33 (1972), 217-233 Zbl0245.26011MR2199372
  25. J. Lott, C. Villani, Ricci curvature for metric-measure spaces via optimal transport Zbl1178.53038
  26. F. Maggi, C. Villani, Balls have the worst best Sobolev inequality Zbl1086.46021
  27. B. Maurey, Some deviation inequalities, Geom. Funct. Anal. 1 (1991), 188-197 Zbl0756.60018MR1097258
  28. B. Maurey, Inégalité de Brunn-Minkowski-Lusternik, et autres inégalités géométriques et fonctionnelles, Séminaire Bourbaki (2003) Zbl1101.52002MR2167203
  29. R.J. McCann, A Convexity Principle for Interacting Gases and Equilibrium Crystals, (1994) 
  30. R.J. McCann, Existence and uniqueness of monotone measure-preserving maps, Duke. Math. J. 80 (1995), 309-323 Zbl0873.28009MR1369395
  31. R.J. McCann, A convexity principle for interacting gases, Adv. Math. 128 (1997), 153-179 Zbl0901.49012MR1451422
  32. R.J. McCann, Polar factorization of maps on Riemannian manifolds, Geom. Funct. Anal. 11 (2001), 589-608 Zbl1011.58009MR1844080
  33. V.D. Milman, G. Schechtman, Asymptotic theory of finite-dimensional normed spaces, (1986), Springer-Verlag, Berlin Zbl0606.46013MR856576
  34. F. Otto, The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations 26 (2001), 101-174 Zbl0984.35089MR1842429
  35. F. Otto, C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal. 173 (2000), 361-400 Zbl0985.58019MR1760620
  36. A. Prékopa, Logarithmic concave measures with application to stochastic programming, Acta Sci. Math. 32 (1971), 301-315 Zbl0235.90044MR315079
  37. A. Prékopa, On logarithmic concave measures and functions, Acta Sci. Math. (Szeged) 34 (1973), 335-343 Zbl0264.90038MR404557
  38. M. Schmuckenschläger, A concentration of measure phenomenon on uniformly convex bodies, GAFA Seminar (1992-1994) (1995), 275-287, Birkaäuser Zbl0828.52004MR1353466
  39. R. Schneider, Convex Bodies: the Brunn-Minkowski Theory, (1993), Cambridge University Press, Cambridge Zbl0798.52001MR1216521
  40. K.-T. Sturm, Convex functionals of probability measures and nonlinear diffusions, J. Math. Pures Appl. 84 (2005) Zbl1259.49074MR2118836
  41. K.-T. Sturm, M.-K. von Renesse, Transport inequalities, gradient estimates, entropy and Ricci curvature, Comm. Pure Appl. Math. 58 (2005), 923-940 Zbl1078.53028MR2142879
  42. N.S. Trudinger, Isoperimetric inequalities for quermassintegrals, Ann. Inst. H. Poincaré Anal. Non Linéaire 11 (1994), 411-425 Zbl0859.52001MR1287239
  43. C. Villani, Graduate Studies in Math., Topics in Optimal Transportation 58 (2003), American Mathematical Society, Providence, RI Zbl1106.90001MR1964483

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.