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38
On the role of convexity in functional and isoperimetric inequalities
 the Proc. London Math. Soc., arxiv.org/abs/0804.0453
, 2008
"... This is a continuation of our previous work [41]. It is well known that various isoperimetric inequalities imply their functional “counterparts”, but in general this is not an equivalence. We show that under certain convexity assumptions (e.g. for logconcave probability measures in Euclidean space) ..."
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This is a continuation of our previous work [41]. It is well known that various isoperimetric inequalities imply their functional “counterparts”, but in general this is not an equivalence. We show that under certain convexity assumptions (e.g. for logconcave probability measures in Euclidean space), the latter implication can in fact be reversed for very general inequalities, generalizing a reverse form of Cheeger’s inequality due to Buser and Ledoux. We develop a coherent single framework for passing between isoperimetric inequalities, OrliczSobolev functional inequalities and capacity inequalities, the latter being notions introduced by Maz’ya and extended by Barthe–Cattiaux–Roberto. As an application, we extend the known results due to the latter authors about the stability of the isoperimetric profile under tensorization, when there is no CentralLimit obstruction. As another application, we show that under our convexity assumptions, qlogSobolev inequalities (q ∈ [1, 2]) are equivalent to an appropriate family of isoperimetric inequalities, extending results of Bakry–Ledoux and Bobkov–Zegarlinski. Our results extend to the more general setting of Riemannian manifolds with density which satisfy the CD(0, ∞) curvaturedimension condition of Bakry – Émery. 1
Functional inequalities for heavy tails distributions and application to isoperimetry
, 2008
"... Abstract. This paper is devoted to the study of probability measures with heavy tails. Using the Lyapunov function approach we prove that such measures satisfy different kind of functional inequalities such as weak Poincaré and weak Cheeger, weighted Poincaré and weighted Cheeger inequalities and th ..."
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Abstract. This paper is devoted to the study of probability measures with heavy tails. Using the Lyapunov function approach we prove that such measures satisfy different kind of functional inequalities such as weak Poincaré and weak Cheeger, weighted Poincaré and weighted Cheeger inequalities and their dual forms. Proofs are short and we cover very large situations. For product measures onR n we obtain the optimal dimension dependence using the mass transportation method. Then we derive (optimal) isoperimetric inequalities. Finally we deal with spherically symmetric measures. We recover and improve many previous results.
An isoperimetric inequality on the ℓp balls
, 2008
"... The normalised volume measure on the ℓ n p unit ball (1 ≤ p ≤ 2) satisfies the following isoperimetric inequality: the boundary measure of a set of measure a is at least cn 1/p ã log 1−1/p (1/ã), where ã = min(a,1 − a). Résumé Nous prouvons une inégalité isopérimétrique pour la mesure uniforme Vp,n ..."
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The normalised volume measure on the ℓ n p unit ball (1 ≤ p ≤ 2) satisfies the following isoperimetric inequality: the boundary measure of a set of measure a is at least cn 1/p ã log 1−1/p (1/ã), where ã = min(a,1 − a). Résumé Nous prouvons une inégalité isopérimétrique pour la mesure uniforme Vp,n sur la boule unité de ℓ n p (1 ≤ p ≤ 2). Si Vp,n(A) = a, alors V + p,n(A) ≥ cn 1/p ea log 1−1/p 1/ea, où V + p,n est la mesure de surface associée à Vp,n, ea = min(a,1 − a) et c est une constante absolue. En particulier, les boules unités de ℓ n p vérifient la conjecture de Kannan– Lovász–Simonovits [KLS] sur la constante de Cheeger d’un corps convexe isotrope. La démonstration s’appuie sur les inégalités isopérimétriques de Bobkov [B1] et de Barthe–Cattiaux–Roberto [BCR], et utilise la représentation de Vp,n établie par Barthe–Guédon–Mendelson–Naor [BGMN] ainsi qu’un argument de découpage. 1
Bernstein type’s concentration inequalities for symmetric Markov processes
"... Abstract. Using the method of transportationinformation inequality introduced in [28],weestablishBernsteintype’sconcentrationinequalitiesforempiricalmeans 1 ∫ t t 0 g(Xs)ds where g is a unbounded observable of the symmetric Markov process (Xt). Three approaches are proposed: functional inequalities ..."
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Abstract. Using the method of transportationinformation inequality introduced in [28],weestablishBernsteintype’sconcentrationinequalitiesforempiricalmeans 1 ∫ t t 0 g(Xs)ds where g is a unbounded observable of the symmetric Markov process (Xt). Three approaches are proposed: functional inequalities approach; Lyapunov function method; and an approach through the Lipschitzian norm of the solution to the Poisson equation. Several applications and examples are studied.
Weak logarithmic Sobolev inequalities and entropic convergence
, 2005
"... In this paper we introduce and study a weakened form of logarithmic Sobolev inequalities in connection with various others functional inequalities (weak Poincaré inequalities, general Beckner inequalities...). We also discuss the quantitative behaviour of relative entropy along a symmetric diffusion ..."
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In this paper we introduce and study a weakened form of logarithmic Sobolev inequalities in connection with various others functional inequalities (weak Poincaré inequalities, general Beckner inequalities...). We also discuss the quantitative behaviour of relative entropy along a symmetric diffusion semigroup. In particular, we exhibit an example where Poincaré inequality can not be used for deriving entropic convergence whence weak logarithmic Sobolev inequality ensures the result.
Hypercontractive measures, Talagrand’s inequality and influences
, 2011
"... Abstract. – We survey several Talagrand type inequalities and their application to influences with the tool of hypercontractivity for both discrete and continuous, and product and nonproduct models. The approach covers similarly by a simple interpolation the framework of geometric influences recent ..."
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Abstract. – We survey several Talagrand type inequalities and their application to influences with the tool of hypercontractivity for both discrete and continuous, and product and nonproduct models. The approach covers similarly by a simple interpolation the framework of geometric influences recently developed by N. Keller, E. Mossel and A. Sen. Geometric BrascampLieb decompositions are also considered in this context. 1.
Convex Sobolev inequalities and spectral gap
, 2005
"... This note is devoted to the proof of convex Sobolev (or generalized Poincaré) inequalities which interpolate between spectral gap (or Poincaré) inequalities and logarithmic Sobolev inequalities. We extend to the whole family of convex Sobolev inequalities results which have recently been obtained by ..."
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This note is devoted to the proof of convex Sobolev (or generalized Poincaré) inequalities which interpolate between spectral gap (or Poincaré) inequalities and logarithmic Sobolev inequalities. We extend to the whole family of convex Sobolev inequalities results which have recently been obtained by Cattiaux [11] and Carlen and Loss [10] for logarithmic Sobolev inequalities. Under local conditions on the density of the measure with respect to a reference measure, we prove that spectral gap inequalities imply all convex Sobolev inequalities with constants which are uniformly bounded in the limit approaching the logarithmic Sobolev inequalities. We recover the case of the logarithmic Sobolev inequalities as a special case.
Modified logSobolev inequalities and isoperimetry
, 2006
"... We find sufficient conditions for a probability measure µ to satisfy an inequality of the type Rd f 2 ( ..."
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We find sufficient conditions for a probability measure µ to satisfy an inequality of the type Rd f 2 (
Poincaré inequalities for non euclidean metrics and . . .
, 2007
"... In this paper, we consider Poincaré inequalities for non euclidean metrics on R d. These inequalities enable us to derive precise dimension free concentration inequalities for product measures. This technique is appropriate for a large scope of concentration rate: between exponential and gaussian ..."
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In this paper, we consider Poincaré inequalities for non euclidean metrics on R d. These inequalities enable us to derive precise dimension free concentration inequalities for product measures. This technique is appropriate for a large scope of concentration rate: between exponential and gaussian and beyond. We give different equivalent functional forms of these Poincaré type inequalities in terms of transportationcost inequalities and infimum convolution inequalities. Workable sufficient conditions are given and a comparison is made with generalized BecknerLatalaOleszkiewicz inequalities.
Isoperimetry and symmetrization for Sobolev spaces on metric spaces
 Comptes Rendus Math
"... Abstract. Using isoperimetry we obtain new symmetrization inequalities that allow us to provide a unified framework to study Sobolev inequalities in metric spaces. The applications include concentration inequalities, as well as metric versions of the PólyaSzegö and FaberKrahn principles. 1. ..."
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Abstract. Using isoperimetry we obtain new symmetrization inequalities that allow us to provide a unified framework to study Sobolev inequalities in metric spaces. The applications include concentration inequalities, as well as metric versions of the PólyaSzegö and FaberKrahn principles. 1.