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183
A masstransportation approach to sharp Sobolev and GagliardoNirenberg inequalities
 MATH
, 2004
"... We show that mass transportation methods provide an elementary and powerful approach to the study of certain functional inequalities with a geometric content, like sharp Sobolev or GagliardoNirenberg inequalities. The Euclidean structure of Rn plays no role in our approach: we establish these inequ ..."
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Cited by 111 (2 self)
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We show that mass transportation methods provide an elementary and powerful approach to the study of certain functional inequalities with a geometric content, like sharp Sobolev or GagliardoNirenberg inequalities. The Euclidean structure of Rn plays no role in our approach: we establish these inequalities, together with cases of equality, for an arbitrary norm.
Solving convex programs by random walks
 Journal of the ACM
, 2002
"... Minimizing a convex function over a convex set in ndimensional space is a basic, general problem with many interesting special cases. Here, we present a simple new algorithm for convex optimization based on sampling by a random walk. It extends naturally to minimizing quasiconvex functions and to ..."
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Cited by 74 (12 self)
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Minimizing a convex function over a convex set in ndimensional space is a basic, general problem with many interesting special cases. Here, we present a simple new algorithm for convex optimization based on sampling by a random walk. It extends naturally to minimizing quasiconvex functions and to other generalizations.
A mass transportation approach to quantitative isoperimetric inequalities
 Invent. Math
, 2010
"... Abstract. A sharp quantitative version of the anisotropic isoperimetric inequality is established, corresponding to a stability estimate for the Wulff shape of a given surface tension energy. This is achieved by exploiting mass transportation theory, especially Gromov’s proof of the isoperimetric ..."
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Cited by 72 (22 self)
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Abstract. A sharp quantitative version of the anisotropic isoperimetric inequality is established, corresponding to a stability estimate for the Wulff shape of a given surface tension energy. This is achieved by exploiting mass transportation theory, especially Gromov’s proof of the isoperimetric inequality and the BrenierMcCann Theorem. A sharp quantitative version of the BrunnMinkowski inequality for convex sets is proved as a corollary. 1.
An introduction to stochastic PDEs
 Lecture notes, 2009. URL http://arxiv.org/abs/0907.4178
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A BrunnMinkowski type inequality for Fano manifolds and the BandoMabuchi uniqueness theorem
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The (B) conjecture for the Gaussian measure of dilates of symmetric convex sets and related problems
 GAUSSIAN MEASURE OF SECTIONS OF DILATES AND TRANSLATIONS. 9
"... We show that given a symmetric convex set K ⊂ Rd, the function t − → γ(etK) is logconcave on R, where γ denotes the standard ddimensional Gaussian measure. We also comment on the extension of this property to unconditional logconcave measures and sets, and on the complex case. 1 ..."
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Cited by 31 (2 self)
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We show that given a symmetric convex set K ⊂ Rd, the function t − → γ(etK) is logconcave on R, where γ denotes the standard ddimensional Gaussian measure. We also comment on the extension of this property to unconditional logconcave measures and sets, and on the complex case. 1
Weighted Poincarétype inequalities for Cauchy and other convex measures
 ANNALS OF PROBABILITY
, 2007
"... Brascamp–Liebtype, weighted Poincarétype and related analytic inequalities are studied for multidimensional Cauchy distributions and more general κconcave probability measures (in the hierarchy of convex measures). In analogy with the limiting (infinitedimensional logconcave) Gaussian model, th ..."
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Cited by 30 (3 self)
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Brascamp–Liebtype, weighted Poincarétype and related analytic inequalities are studied for multidimensional Cauchy distributions and more general κconcave probability measures (in the hierarchy of convex measures). In analogy with the limiting (infinitedimensional logconcave) Gaussian model, the weighted inequalities fully describe the measure concentration and large deviation properties of this family of measures. Cheegertype isoperimetric inequalities are investigated similarly, giving rise to a common weight in the class of concave probability measures under consideration.
Interpolating thinshell and sharp largedeviation estimates for isotropic logconcave measures
 Geom. Funct. Anal
"... Abstract. Given an isotropic random vector X with logconcave density in Euclidean space Rn, we study the concentration properties of X  on all scales, both above and below its expectation. We show in particular that P (∣∣X  − √n∣ ∣ ≥ t√n) ≤ C exp(−cn1/2 min(t3, t)) ∀t ≥ 0, for some universa ..."
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Cited by 26 (4 self)
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Abstract. Given an isotropic random vector X with logconcave density in Euclidean space Rn, we study the concentration properties of X  on all scales, both above and below its expectation. We show in particular that P (∣∣X  − √n∣ ∣ ≥ t√n) ≤ C exp(−cn1/2 min(t3, t)) ∀t ≥ 0, for some universal constants c, C> 0. This improves the best known deviation results on the thinshell and mesoscopic scales due to Fleury and Klartag, respectively, and recovers the sharp largedeviation estimate of Paouris. Another new feature of our estimate is that it improves when X is ψα (α ∈ (1, 2]), in precise agreement with Paouris ’ estimates. The upper bound on the thinshell width Var(X) we obtain is of the order of n1/3, and improves down to n1/4 when X is ψ2. Our estimates thus continuously interpolate between a new best known thinshell estimate and the sharp largedeviation estimate of Paouris. As a consequence, a new best known bound on the Cheeger isoperimetric constant appearing in a conjecture of Kannan–Lovász– Simonovits is deduced. 1
Geometric Inequalities via a General Comparison . . .
, 2003
"... The article builds on several recent advances in the MongeKantorovich theory of mass transport which have  among other things  led to new and quite natural proofs for a wide range of geometric inequalities such as the ones formulated by BrunnMinkowski, Sobolev, GagliardoNirenberg, Beckner, Gr ..."
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Cited by 25 (6 self)
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The article builds on several recent advances in the MongeKantorovich theory of mass transport which have  among other things  led to new and quite natural proofs for a wide range of geometric inequalities such as the ones formulated by BrunnMinkowski, Sobolev, GagliardoNirenberg, Beckner, Gross, Talagrand, OttoVillani and their extensions by many others. While this paper continues in this spirit, we however propose here a basic framework to which all of these inequalities belong, and a general unifying principle from which many of them follow. This basic inequality relates the relative total energy  internal, potential and interactive  of two arbitrary probability densities, their Wasserstein distance, their barycenters and their entropy production functional. The framework is remarkably encompassing as it implies many old geometric  Gaussian and Euclidean  inequalities as well as new ones, while allowing a direct and unified way for computing best constants and extremals. As expected, such inequalities also lead to exponential rates of convergence to equilibria for solutions of FokkerPlanck and McKeanVlasov type equations. The principle also leads to a remarkable correspondence between ground state solutions of certain quasilinear  or semilinear  equations and stationary solutions of  nonlinear  FokkerPlanck type equations.