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Generalization Of An Inequality By Talagrand, And Links With The Logarithmic Sobolev Inequality
 J. Funct. Anal
, 2000
"... . We show that transport inequalities, similar to the one derived by Talagrand [30] for the Gaussian measure, are implied by logarithmic Sobolev inequalities. Conversely, Talagrand's inequality implies a logarithmic Sobolev inequality if the density of the measure is approximately logconcave, in a ..."
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Cited by 121 (10 self)
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. We show that transport inequalities, similar to the one derived by Talagrand [30] for the Gaussian measure, are implied by logarithmic Sobolev inequalities. Conversely, Talagrand's inequality implies a logarithmic Sobolev inequality if the density of the measure is approximately logconcave, in a precise sense. All constants are independent of the dimension, and optimal in certain cases. The proofs are based on partial dierential equations, and an interpolation inequality involving the Wasserstein distance, the entropy functional and the Fisher information. Contents 1. Introduction 1 2. Main results 5 3. Heuristics 11 4. Proof of Theorem 1 18 5. Proof of Theorem 3 24 6. An application of Theorem 1 30 7. Linearizations 31 Appendix A. A nonlinear approximation argument 35 References 36 1. Introduction Let M be a smooth complete Riemannian manifold of dimension n, with the geodesic distance d(x; y) = inf 8 < : s Z 1 0 j _ w(t)j 2 dt; w 2 C 1 ((0; 1); M); w(0) = x; w(1) = y 9 ...
An Elementary Introduction to Modern Convex Geometry
 in Flavors of Geometry
, 1997
"... Introduction to Modern Convex Geometry KEITH BALL Contents Preface 1 Lecture 1. Basic Notions 2 Lecture 2. Spherical Sections of the Cube 8 Lecture 3. Fritz John's Theorem 13 Lecture 4. Volume Ratios and Spherical Sections of the Octahedron 19 Lecture 5. The BrunnMinkowski Inequality and Its Ext ..."
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Cited by 99 (2 self)
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Introduction to Modern Convex Geometry KEITH BALL Contents Preface 1 Lecture 1. Basic Notions 2 Lecture 2. Spherical Sections of the Cube 8 Lecture 3. Fritz John's Theorem 13 Lecture 4. Volume Ratios and Spherical Sections of the Octahedron 19 Lecture 5. The BrunnMinkowski Inequality and Its Extensions 25 Lecture 6. Convolutions and Volume Ratios: The Reverse Isoperimetric Problem 32 Lecture 7. The Central Limit Theorem and Large Deviation Inequalities 37 Lecture 8. Concentration of Measure in Geometry 41 Lecture 9. Dvoretzky's Theorem 47 Acknowledgements 53 References 53 Index 55 Preface These notes are based, somewhat loosely, on three series of lectures given by myself, J. Lindenstrauss and G. Schechtman, during the Introductory Workshop in Convex Geometry held at the Mathematical Sciences Research Institute in Berkeley, early in 1996. A fourth series was given by B. Bollobas, on rapid mixing and random volume algorithms; they are found els
Exponential integrability and transportation cost related to logarithmic Sobolev inequalities
 J. Funct. Anal
, 1999
"... We study some problems on exponential integrability, concentration of measure, and transportation cost related to logarithmic Sobolev inequalities. On the real line, we then give a characterization of those probability measures which satisfy these inequalities 1999 Academic Press Key Words: logarith ..."
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Cited by 80 (4 self)
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We study some problems on exponential integrability, concentration of measure, and transportation cost related to logarithmic Sobolev inequalities. On the real line, we then give a characterization of those probability measures which satisfy these inequalities 1999 Academic Press Key Words: logarithmic Sobolev inequalities; exponential integrability; concentration of measure; transportation inequalities.
The BrunnMinkowski inequality
 Bull. Amer. Math. Soc. (N.S
, 2002
"... Abstract. In 1978, Osserman [124] wrote an extensive survey on the isoperimetric inequality. The BrunnMinkowski inequality can be proved in a page, yet quickly yields the classical isoperimetric inequality for important classes of subsets of R n, and deserves to be better known. This guide explains ..."
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Cited by 74 (5 self)
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Abstract. In 1978, Osserman [124] wrote an extensive survey on the isoperimetric inequality. The BrunnMinkowski inequality can be proved in a page, yet quickly yields the classical isoperimetric inequality for important classes of subsets of R n, and deserves to be better known. This guide explains the relationship between the BrunnMinkowski inequality and other inequalities in geometry and analysis, and some applications. 1.
A Riemannian interpolation inequality à la Borell, Brascamp and Lieb
, 2001
"... A concavity estimate is derived for interpolations between L¹(M) mass densities on a Riemannian manifold. The inequality sheds new light on the theorems of Prékopa, Leindler, Borell, Brascamp and Lieb that it generalizes from Euclidean space. Due to the curvature of the manifold, the new Riemannian ..."
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Cited by 56 (7 self)
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A concavity estimate is derived for interpolations between L¹(M) mass densities on a Riemannian manifold. The inequality sheds new light on the theorems of Prékopa, Leindler, Borell, Brascamp and Lieb that it generalizes from Euclidean space. Due to the curvature of the manifold, the new Riemannian versions of these theorems incorporate a volume distortion factor which can, however, be controlled via lower bounds on Ricci curvature. The method uses optimal mappings from mass transportation theory. Along the way, several new properties are established for optimal mass transport and interpolating maps on a Riemannian manifold.
Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry
, 2004
"... ..."
Isoperimetric Constants For Product Probability Measures
 Ann. Probab
, 1995
"... A dimension free lower bound is found for isoperimetric constants of product probability measures. From this, some analytic inequalities are derived. 1 Introduction Let (X; d) be a metric space equipped with a nonatomic, separable Borel probability measure ¯. In the present paper we study the qua ..."
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Cited by 28 (5 self)
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A dimension free lower bound is found for isoperimetric constants of product probability measures. From this, some analytic inequalities are derived. 1 Introduction Let (X; d) be a metric space equipped with a nonatomic, separable Borel probability measure ¯. In the present paper we study the quantity Is(¯) = inf ¯ + (A) min(¯(A); 1 \Gamma ¯(A)) (1:1) Key words: Isoperimetry, Poincar'e Inequalities, Cheeger's inequality, Khinchine Kahane inequality, Holder's Inequality y Research supported in part by the ISF grant NZX000 and NZX300. This author enjoyed the hospitality of the Faculty of Wiskunde and Informatica, Free University of Amsterdam, while part of this research was carried out. z Research supported in part by an NSF Postdoctoral Fellowship. This author enjoyed the hospitality of Le Cermics, ENPC, France, of the Steklov Mathematical Institute (Sankt Petersburg branch) and of the Department of Mathematics, University of Syktyvkar, Russia, while part of this research...
Inequalities for generalized entropy and optimal transportation
 IN PROCEEDINGS OF THE WORKSHOP: MASS TRANSPORTATION METHODS IN KINETIC THEORY AND HYDRODYNAMICS
, 2003
"... A new concept of Fisherinformation is introduced through a cost function. That concept is used to obtain extensions and variants of transport and logarithmic Sobolev inequalities for general entropy functionals and transport costs. Our proofs rely on optimal mass transport from the MongeKantorovic ..."
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Cited by 21 (3 self)
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A new concept of Fisherinformation is introduced through a cost function. That concept is used to obtain extensions and variants of transport and logarithmic Sobolev inequalities for general entropy functionals and transport costs. Our proofs rely on optimal mass transport from the MongeKantorovich theory. They express the convexity of entropy functionals with respect to suitably chosen paths on the set of probability measures.
Characterization of Talagrand’s like transportationcost inequalities on the real line
, 2006
"... In this paper, we give necessary and sufficient conditions for Talagrand’s like transportation cost inequalities on the real line. This brings a new wide class of examples of probability measures enjoying a dimensionfree concentration of measure property. Another byproduct is the characterization ..."
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Cited by 14 (4 self)
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In this paper, we give necessary and sufficient conditions for Talagrand’s like transportation cost inequalities on the real line. This brings a new wide class of examples of probability measures enjoying a dimensionfree concentration of measure property. Another byproduct is the characterization of modified LogSobolev inequalities for Logconcave probability measures on R.
Prékopa–Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport
, 2005
"... We investigate PrékopaLeindler type inequalities on a Riemannian manifold M equipped with a measure with density e−V where the potential V and the Ricci curvature satisfy Hessx V + Ricx ≥ λ I for all x ∈ M, with some λ ∈ R. As in our earlier work [14], the argument uses optimal mass transport on M, ..."
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Cited by 10 (2 self)
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We investigate PrékopaLeindler type inequalities on a Riemannian manifold M equipped with a measure with density e−V where the potential V and the Ricci curvature satisfy Hessx V + Ricx ≥ 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 BakryEmery criterion will be recovered) and to transport inequalities. A study of the displacement convexity of the entropy functional completes the exposition.