From Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities

Geometric and Functional Analysis

Volumn 10, Issue 5, 2000, Pages 1028-1052

  Bobkov, S G ^a   Ledoux, M ^b  

^a UNIVERSITY OF MINNESOTA   (United States)

^b INSTITUT DE MATHÉMATIQUES DE TOULOUSE   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034553975     PISSN: 1016443X     EISSN: None     Source Type: Journal    
DOI: 10.1007/PL00001645     Document Type: Article

References (31)

1. S. AIDA, Uniform positivity improving property, Sobolev inequalities and spectral gaps, J. Funct. Anal. 158 (1998), 152-185.
2. S. AIDA, T. MASUDA, I. SHIGEKAWA, Logarithmic Sobolev inequalities and exponential integrability, J. Funct. Anal. 126 (1994), 83-101.
3. J. ARIAS-DE-REYNA, K. BALL, R. VILLA, Concentration of the distance in finite dimensional normed spaces, preprint (1998).
4. D. BAKRY, L'hypercontractivité et son utilisation en théorie des semi-groupes, Ecole d'Eté de Probabilités de St-Flour, Springer Lecture Notes in Math. 1581 (1994), 1-114.
5. D. BAKRY, M. EMERY, Diffusions hypercontractives, Séminaire de Probabilités XIX, Springer Lecture Notes in Math. 1123 (1985), 177-206.
6. W. BECKNER, Geometric asymptotics and the logarithmic Sobolev inequality, preprint (1998).
7. G. BLOWER, The Gaussian isoperimetric inequality and transportation, preprint (1999).
8. S. BOBKOV, Isoperimetric and analytic inequalities for log-concave probability measures (1998), Ann. Probability 27:4 (1999), 1903-1921.
9. S. BOBKOV, F. GÖTZE, Exponential integrability and transportation cost related to logarithmic Sobolev inequalities, J. Funct. Anal. 163 (1999), 1-28.
10. 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, 366-389 (1976).
11. Y. BRENIER, Polar factorization and monotone rearrangement of vector-valued functions, Comm. Pure Appl. Math. 44 (1991), 375-417.
12. A. DEMBO, Information inequalities and concentration of measure, Ann. Probability 25 (1997), 927-939.
13. M. GROMOV, V.D. MILMAN, Generalization of the spherical isoperimetric inequality to uniformly convex Banach spaces, Compositio Math. 62 (1987), 263-282.
14. L. GROSS, Logarithmic Sobolev inequalities, Amer. J. Math. 97 (1975), 1061-1083.
15. H.G. KELLERER, Duality theory for marginal problems, Z. Wahrscheinlichkeitstheorie verw. Gebiete 67 (1984), 399-482.
16. M. LEDOUX, Concentration of measure and logarithmic Sobolev inequalities, Séminaire de Probabilités XXXIII, Springer Lecture Notes in Math., to appear.
17. L. LEINDLER, On a certain converse of Hölder's inequality II, stochastic programming, Acta Sci. Math. Szeged 33 (1972), 217-223.
18. V.L. LEVIN, The problem of mass transfer in a topological space, and probability measures having given marginal measures on the product of two spaces, Soviet Math. Dokl. 29:3 (1984), 638-643.
19. J. LINDENSTRAUSS, L. TZAFRIRI, Classical Banach Spaces II, Springer, 1979.
20. K. MARTON, A measure concentration inequality for contracting Markov chains, Geometric And Funct. Anal. 6 (1997), 556-571.
21. B. MAUREY, Some deviations inequalities, Geometric And Funct. Anal. 1 (1991), 188-197.
22. R.J. MCCANN, Existence and uniqueness of monotone measure-preserving maps, Duke Math. J. 80 (1995), 309-323.
23. F. OTTO, C. VILLANI, Generalization of an inequality by Talagrand, and links with the logarithmic Sobolev inequality, preprint (1999).
24. G. PISIER, Martingales with values in uniformly convex spaces, Israel J. Math. 20 (1975), 326-350.
25. G. PISIER, The Volume of Convex Bodies and Banach Space Geometry, Cambridge Univ. Press, 1989.
26. A. PRÉKOPA, Logarithmic concave measures with applications to stochastic programming, Acta Sci. Math. Szeged 32 (1971), 301-316.
27. A. PRÉKOPA, On logarithmic concave measures and functions, Acta Sci. Math. Szeged 34 (1973), 335-343.
28. S.T. RACHEV, The Monge-Kantorovich mass transference problem and its stochastic applications, Theory Probab. Appl. 24 (1984), 647-671.
29. M. SCHMUCKENSCHLÄGER, A concentration of measure phenomenon on uniformly convex bodies, Geometric Aspects of Functional Analysis (Israel 1992-94), Birkhäuser, Oper. Theory Adv. Appl. 77 (1995), 275-287.
30. M. TALAGRAND, Transportation cost for Gaussian and other product measures, Geometric And Funct. Anal. 6 (1996), 587-600.
31. F.-Y. WANG, Logarithmic Sobolev inequalities on noncompact Riemannian manifolds, Probab. Theor. Relat. Fields 109 (1997), 417-424.