Quelques exemples d'application du transport de mesure en géométrie euclidienne et riemannienne

Dario Cordero-Erausquin

Séminaire de théorie spectrale et géométrie (2003-2004)

  • Volume: 22, page 125-152
  • ISSN: 1624-5458

How to cite

top

Cordero-Erausquin, Dario. "Quelques exemples d'application du transport de mesure en géométrie euclidienne et riemannienne." Séminaire de théorie spectrale et géométrie 22 (2003-2004): 125-152. <http://eudml.org/doc/114481>.

@article{Cordero2003-2004,
author = {Cordero-Erausquin, Dario},
journal = {Séminaire de théorie spectrale et géométrie},
keywords = {Optimal transport; geometric inequality; Prekopa-Leindler inequality},
language = {fre},
pages = {125-152},
publisher = {Institut Fourier},
title = {Quelques exemples d'application du transport de mesure en géométrie euclidienne et riemannienne},
url = {http://eudml.org/doc/114481},
volume = {22},
year = {2003-2004},
}

TY - JOUR
AU - Cordero-Erausquin, Dario
TI - Quelques exemples d'application du transport de mesure en géométrie euclidienne et riemannienne
JO - Séminaire de théorie spectrale et géométrie
PY - 2003-2004
PB - Institut Fourier
VL - 22
SP - 125
EP - 152
LA - fre
KW - Optimal transport; geometric inequality; Prekopa-Leindler inequality
UR - http://eudml.org/doc/114481
ER -

References

top
  1. [1] M. AGUEH, N. GHOUSSOUB ET X. KANG, Geometric inequalities via a general comparison principlefor interacting gases, Geom. Funct. Anal., 14 ( 2004), 215-244. Zbl1122.82022MR2053603
  2. [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. Zbl0877.35040MR1441395
  3. [3] T. AUBIN, Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geometry 11,4 ( 1976), 573-598. Zbl0371.46011MR448404
  4. [4] D. BAKRY, ET M. ÉMERYDiffusions hypercontractives, in Séminaire de Probabilités XIX, Lecture Notes in Math., 1123, Springer ( 1985), 177-206. Zbl0561.60080MR889476
  5. [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. Zbl0901.52002
  6. [6] F. BARTHE, On a reverse form of the Brascamp-Lieb inequality, Invent. Math., 134 ( 1998), n°2, 335-361. Zbl0901.26010MR1650312
  7. [7] F. BARTHE, Optimal Young's inequality and its converse : a simple proof, Geom. Funct. Anal, 8 ( 1998), 234-242. Zbl0902.26009MR1616143
  8. [8] S. BOBKOV, ET M. LEDOUXFrom Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities, Geom. Funct. Anal., 10 ( 2000), 1028-1052. Zbl0969.26019MR1800062
  9. [9] S. BOBKOV, I. GENTIL ET M. LEDOUX, Hypercontractivity of Hamilton-Jacobi equations, J. Math. Pures Appl (9) 80 ( 2001), no4, 669-696. Zbl1038.35020MR1846020
  10. [10] C. BORELL, Convex set functions in d-space, Period. Math. Hungar.,6 ( 1975), 111-136. Zbl0274.28009MR404559
  11. [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. Zbl0334.26009MR450480
  12. [12] Y. BRENIER, Décomposition polaire et réarrangement monotone des champs de vecteurs, C.R. Acad. Sci. Paris Sér. I Math., 305 ( 1987), 805-808. Zbl0652.26017MR923203
  13. [13] Y. BRENIER, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl Math., 44 ( 1991), 375-417. Zbl0738.46011MR1100809
  14. [14] J. BROTHERS ET W. ZIEMERMinimal rearrangements of Sobolev functions, J. Reine Angew. Math., 384 ( 1988), 153-179. Zbl0633.46030MR929981
  15. [15] D. CORDERO -ERAUSQUIN, Some applications of mass transport to Gaussian type inequalities, Arch. Rational Mech. Anal., 161 ( 2002), 257-269. Zbl0998.60080MR1894593
  16. [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. Zbl1135.49026MR2079071
  17. [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. Zbl1026.58018MR1865396
  18. [18] D. CORDERO -ERAUSQUIN, RJ. MCCANN ET M. SCHMUCKENSCHLÄGER, Prékopa-Leindler type inequalities on Riemannian manifolds, mass transport and Jacobi fields, Preprint ( 2004). Zbl1125.58007MR2079070
  19. [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. Zbl1048.26010MR2032031
  20. [20] S. DAS GUPTA, Brunn-Minkowski inequality and its aftermath, J. Multivariate Anal., 10 ( 1980), 296-318. Zbl0467.26008MR588074
  21. [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. Zbl1023.58009MR1938183
  22. [22] S. GALLOT, D. HULIN ET J. LAFONTAINE, Riemannian Geometry, Springer-Verlag (Universitext), Berlin, 1990. Zbl0716.53001MR1083149
  23. [23] RJ. GARDNER, The Brunn-Minkowski inequality, Bull. Amer. Math. Soc. (N.S.), 39-3 ( 2002), 355-405. Zbl1019.26008MR1898210
  24. [24] M. GROMOV ET V. MILMAN, A topological application of the isoperimetric inequatliry, Amer. J. Math., 105 ( 1983), 843-854. Zbl0522.53039MR708367
  25. [25] H. HADWIGER ET D. OHMANN, Brunn-Minkowskischer Satz und Isoperimetrie, Math. Z, 66 ( 1956), 1-8. Zbl0071.38001MR82697
  26. [26] H. KNOTHE, Contributions to the theory of convex bodies, Michigan Math, J., 4 ( 1957), 39-52. Zbl0077.35803MR83759
  27. [27] M. LEDOUX, Concentration of measure and logarithmic Sobolev inequalities, in Séminaire de Probabilités XXXIII, Lecture Notes in Math. 1709, Springer ( 1999), 120-216. Zbl0957.60016MR1767995
  28. [28] M. LEDOUX, The geometry of Markov diffusion generators, Ann. Fac. Sci. Toulouse Math. (6)9 ( 2000), n°2, 305-366. Zbl0980.60097MR1813804
  29. [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. [30] M. LEDOUX, The concentration of measure phenomenon. American Mathematical Society, Providence, RI, 2001. Zbl0995.60002MR1849347
  31. [31] L. LEINDLER, On a certain converse of Hölder's inequality, Acta Sci. Math., 33 ( 1972), 217-233. Zbl0245.26011MR2199372
  32. [32] F. MAGGI ET C. VILLANI, Balls have the worst best Sobolev inequality, Preprint ( 2004). Zbl1086.46021
  33. [33] B. MAUREY, Some deviation inequalities, Geom.Funct. Anal., 1 ( 1991), 188-197. Zbl0756.60018MR1097258
  34. [34] B. MAUREY, Inégalité de Brunn-Minkowski-Lusternik, et autres inégalités géométriques et fonctionnelles, Séminaire Bourbaki, Novembre 2003. Zbl1101.52002MR2167203
  35. [35] R.J. MCCANN, Existence and uniqueness of monotone measure-preserving maps, Duke. Math. J., 80 ( 1995), 309-323. Zbl0873.28009MR1369395
  36. [36] R.J. MCCANN, A convexity principle for interacting gases, Adv. Math., 128 ( 1997), 153-179. Zbl0901.49012MR1451422
  37. [37] R.J. MCCANN, Polar factorizatîon of maps on Riemannian manifolds, Geom. Funct. Anal., 11 ( 2001), n°3, 589-608. Zbl1011.58009MR1844080
  38. [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. Zbl0606.46013MR856576
  39. [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. [40] F. OTTO, The geometry of dissipative evolution equations : the porous medium equation, Comm. Partial Differential Equations, 26 ( 2001), n° 1-2,101-174. Zbl0984.35089MR1842429
  41. [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. Zbl0985.58019MR1760620
  42. [42] A. PRÉKOPA, Logarithmic concave measures with application to stochastic programming, Acta Sci. Math., 32 ( 1971), 301-315. Zbl0235.90044MR315079
  43. [43] A. PRÉKOPA, On logarithmic concave measures and functions, Acta Sci. Math. (Szeged), 34 ( 1973), 335-343. Zbl0264.90038MR404557
  44. [44] R. SCHNEIDER, Convex Bodies : the Brunn-Minkowski Theory, Cambridge University Press, Cambridge, 1993. Zbl0798.52001MR1216521
  45. [45] K.-T. STURM ET M.-K. VON RENESSE, Transport inequalities, gradient estimates, entropy and Ricci curvature, à paraître dans Comm. Pure Appl. Math. Zbl1078.53028MR2142879
  46. [46] G. TALENTI, Best constants in Sobolev inequality, Ann. Mat Pura Appl. (IV), 110 ( 1976), 353-372. Zbl0353.46018MR463908
  47. [47] C. VILLANI, Topicsin Optimal Transportation, Graduate Studies in Math. 58, American Mathematical Society, Providence, RI, 2003. 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.