Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-14T05:21:41.120Z Has data issue: false hasContentIssue false
  • English
  • Français

Transport Inequalities for Log-concave Measures, Quantitative Forms, and Applications

Published online by Cambridge University Press:  20 November 2018

Dario Cordero-Erausquin
Dario Cordero-Erausquin*
Affiliation:
Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie -Paris 6, 75252 Paris Cedex 05, France e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We review some simple techniques based on monotone-mass transport that allow us to obtain transport-type inequalities for any log-concave probability measures, and for more general measures as well. We discuss quantitative forms of these inequalities, with application to the Brascamp–Lieb variance inequality.

Keywords

52A4060E1549Q20log-concave measurestransport inequalityBrascamp-Lieb inequalityquantitative inequalities
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 69 , Issue 3 , 01 June 2017 , pp. 481 - 501
Copyright
Copyright © Canadian Mathematical Society 2017

References

[1] Barthe, F. and Cordero-Erausquin, D., Invariances in variance estimates. Proc. Lond. Math. Soc. (3) 106(2013), 33–64.http://dx.doi.Org/10.1112/plms/pds011 Google Scholar
[2] Barthe, F. and Kolesnikov, A. V. , Mass transport and variants of the logarithmic Sobolev inequality. J. Geom. Anal. 18(2008), no. 4, 921–979. http://dx.doi.org/10.1007/s12220-008-9039-6 Google Scholar
[3] Blower, G., The Gaussian isoperimetric inequality and transportation. Positivity 7(2003), 203–224. http://dx.doi.Org/10.1023/A:1026242611940 Google Scholar
[4] Bobkov, S.G., Gozlan, N., Roberto, C., and Samson, P.-M., Bounds on the deficit in the logarithmic Sobolev inequality. J. Funct. Anal. 267(2014), no. 11, 4110–4138.http://dx.doi.Org/10.1016/j.jfa.2O14.09.016 Google Scholar
[5] Bobkov, S. G. and Houdré, C., Some connections between isoperimetric and Sobolev-type inequalities. Mem. Amer. Math. Soc. 129(1997), no. 616.http://dx.doi.Org/10.1090/memo/0616 Google Scholar
[6] Bobkov, S. G. and Ledoux, M., From Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities. Geom. Funct. Anal. 10(2000), no. 5,1028–1052. http://dx.doi.Org/10.1007/PL00001645 Google Scholar
[7] Bolley, F., Gentil, I., and Guillin, A., Dimensional improvements of the logarithmic Sobolev, Talagrand and Brascamp-Lieb inequalities. arxiv:1 507.01086 Google Scholar
[8] Brascamp, H. J. and 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. 22(1976), no. 4, 366–389.http://dx.doi.Org/10.1016/0022-1236(76)90004-5 Google Scholar
[9] Brenier, Y., Polar factorization and monotone rearrangement of vector-valued functions. Comm. Pure Appl. Math. 44(1991), no. 4, 375–417. http://dx.doi.Org/10.1 OO2/cpa.3160440402 Google Scholar
[10] Cordero-Erausquin, D., Some applications of mass transport to Gaussian-type inequalities. Arch. Ration. Mech. Anal. 161(2002), no. 3, 257–269. http://dx.doi.Org/10.1007/s002050100185 Google Scholar
[11] Cordero-Erausquin, D. and Gozlan, N., Transport proofs of weighted Poincaré inequalities for log-concave distributions. Bernouilli 23(2017), no. 1,134–158.http://dx.doi.Org/10.3150/15-BEJ739 Google Scholar
[12] Cordero-Erausquin, D. and Klartag, B., Moment measures. J. Funct. Anal. 268(2015), no. 12, 3834–3866.http://dx.doi.Org/10.1016/j.jfa.2O15.04.001 Google Scholar
[13] Evans, L. C. and Gariepy, R. F., Measure theory and fine properties of functions. Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.Google Scholar
[14] Fathi, M., Indrei, E., and Ledoux, M., Quantitative logarithmic Sobolev inequalities and stability estimates. Discrete Contin. Dyn. Syst. 36(2016), no. 12, 6835–6853. http://dx.doi.Org/1O.3934/dcds.2O16097 Google Scholar
[15] Figalli, A., Maggi, F., and Pratelli, A., A refined Brunn-Minkowski inequality for convex sets.Ann.Inst. H. Poincaré Anal. Non Linéaire 26(2009), 2511–2519.http://dx.doi.Org/10.1016/j.anihpc.2009.07.004 Google Scholar
[16] De Philippis, G. and Figalli, A., The Monge-Amépre equation and its link to optimal transportation. Bull. Amer. Math. Soc. (N.S.) 51(2014), no. 4, 527–580.http://dx.doi.Org/10.1090/S0273-0979-2014-01459-4 Google Scholar
[17] Gentil, I., From the Prékopa-Leindler inequality to modified logarithmic Sobolev inequality. Ann. Fac. Sci. Toulouse Math. (6) 17(2008), 291–308.http://dx.doi.Org/10.58O2/afst.1184 Google Scholar
[18] Gozlan, N. and Léonard, C., Transport inequalities. A survey. Markov Process. Related Fields 16(2010), no. 4, 635–736.Google Scholar
[19] G.|Hargé, Reinforcement of an inequality due to Brascamp and Lieb. J. Funct. Anal. 254 (2008), no. 2, 267–300.http://dx.doi.Org/10.1016/j.jfa.2007.07.019 Google Scholar
[20] Klartag, B., Concentration of measures supported on the cube. Israel J. Math. 203 (2014), no. 1, 59–80.http://dx.doi.org/10.1007/s11856-013-0072-1 Google Scholar
[21] Ledoux, M., The concentration of measure phenomenon. Mathematical Surveys and Monographs, 89, American Mathematical Society, Providence, RI, 2001.Google Scholar
[22] Ledoux, M., Spectral gap, logarithmic Sobolev constant, and geometric bounds. In: Surveys in differential Geometry, IX, Int. Press, Somerville, MA, 2004, pp. 219–240.http://dx.doi.org/10.4310/SDC.2004.v9.n1.a6 Google Scholar
[23] McCann, R. J., Existence and uniqueness of monotone measure-preserving maps. Duke Math. J. 80(1995), 309–323. http://dx.doi.Org/10.1215/S0012-7094-95-08013-2 Google Scholar
[24] McCann, R. J., A convexity principle for interacting gases. Adv. Math. 128(1997), no. 1,153–179. http://dx.doi.Org/10.1006/aima.1997.1634 Google Scholar
[25] Milman, E., On the role of convexity in isoperimetry, spectral gap and concentration. Invent. Math. 177 (2009), no. 1, 1–43. http://dx.doi.org/10.1OO7/sOO222-OO9-O1 75-9 Google Scholar
[26] Milman, E. and A. Kolesnikov, V., Poincaré and Brunn-Minkowski inequalities on weighted Riemannian manifolds with boundary. arxiv:1310.2526 Google Scholar
[27] Mooney, C., Partial regularity for singular solutions to the Monge-Ampére equation. Comm. Pure Appl. Math. 68(2015), no. 6, 1066–1084.http://dx.doi.Org/10.1 OO2/cpa.21534 Google Scholar
[28] Otto, F. and Villani, C., Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173(2000), 361–400.http://dx.doi.Org/10.1006/jfan.1999.3557 Google Scholar
[29] Rockafellar, R. T., Convex analysis. Princeton Mathematical Series, 28, Princeton University Press, Princeton, NJ, 1970.Google Scholar
[30] Talagrand, M., Transportation cost for Gaussian and other product measures. Geom. Funct. Anal. 6(1996), 587–600.http://dx.doi.org/10.1007/BF02249265 Google Scholar
[31] Veysseire, L., A harmonic mean bound for the spectral gap of the Laplacian on Riemannian manifolds. C. R. Math. Acad. Sci. Paris 348(2010), 1319–1322.http://dx.doi.Org/10.1016/j.crma.2O10.10.015 Google Scholar
[32] Villani, C., Optimal transport. Old and new. Springer-Verlag, Berlin, 2009.Google Scholar