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19
Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry
, 2004
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On the role of convexity in isoperimetry, spectralgap and concentration
 Invent. Math
"... We show that for convex domains in Euclidean space, Cheeger’s isoperimetric inequality, spectral gap of the Neumann Laplacian, exponential concentration of Lipschitz functions, and the apriori weakest requirement that Lipschitz functions have arbitrarily slow uniform taildecay, are all quantitativ ..."
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Cited by 44 (12 self)
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We show that for convex domains in Euclidean space, Cheeger’s isoperimetric inequality, spectral gap of the Neumann Laplacian, exponential concentration of Lipschitz functions, and the apriori weakest requirement that Lipschitz functions have arbitrarily slow uniform taildecay, are all quantitatively equivalent (to within universal constants, independent of the dimension). This substantially extends previous results of Maz’ya, Cheeger, Gromov– Milman, Buser and Ledoux. As an application, we conclude a sharp quantitative stability result for the spectral gap of convex domains under convex perturbations which preserve volume (up to constants) and under maps which are “onaverage ” Lipschitz. We also provide a new characterization (up to constants) of the spectral gap of a convex domain, as one over the square of the average distance from the “worst ” subset having half the measure of the domain. In addition, we easily recover and extend many previously known lower bounds on the spectral gap of convex domains, due to Payne–Weinberger, Li–Yau, Kannan– Lovász–Simonovits, Bobkov and Sodin. The proof involves estimates on the diffusion semigroup following Bakry–Ledoux and a result from Riemannian Geometry on the concavity of the isoperimetric profile. Our results extend to the more general setting of Riemannian manifolds with density which satisfy the CD(0, ∞) curvaturedimension condition of BakryÉmery. 1
Large deviations and isoperimetry over convex probability measures with heavy tails
 Electron J. Prob
, 2007
"... Large deviations and isoperimetric inequalities are considered for probability distributions,
satisfying convexity conditions of the BrunnMinkowskitype. ..."
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Cited by 26 (4 self)
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Large deviations and isoperimetric inequalities are considered for probability distributions,
satisfying convexity conditions of the BrunnMinkowskitype.
Isoperimetry between exponential and Gaussian
 ELECTRONIC J. PROB
, 2008
"... We study in details the isoperimetric profile of product probability measures with tails between the exponential and the Gaussian regime. In particular we exhibit many examples where coordinate halfspaces are approximate solutions of the isoperimetric problem. ..."
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Cited by 23 (8 self)
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We study in details the isoperimetric profile of product probability measures with tails between the exponential and the Gaussian regime. In particular we exhibit many examples where coordinate halfspaces are approximate solutions of the isoperimetric problem.
An isoperimetric inequality for uniformly logconcave measures and uniformly convex bodies
, 2008
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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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Cited by 17 (6 self)
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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
POINTWISE SYMMETRIZATION INEQUALITIES FOR SOBOLEV FUNCTIONS AND APPLICATIONS
, 2009
"... We develop a technique to obtain new symmetrization inequalities that provide a unified framework to study Sobolev inequalities, concentration inequalities and sharp integrability of solutions of elliptic equations. ..."
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Cited by 9 (3 self)
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We develop a technique to obtain new symmetrization inequalities that provide a unified framework to study Sobolev inequalities, concentration inequalities and sharp integrability of solutions of elliptic equations.
Properties of Isoperimetric, Functional and TransportEntropy Inequalities Via Concentration
, 2009
"... Various properties of isoperimetric, functional, TransportEntropy and concentration inequalities are studied on a Riemannian manifold equipped with a measure, whose generalized Ricci curvature is bounded from below. First, stability of these inequalities with respect to perturbation of the measure ..."
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Cited by 7 (3 self)
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Various properties of isoperimetric, functional, TransportEntropy and concentration inequalities are studied on a Riemannian manifold equipped with a measure, whose generalized Ricci curvature is bounded from below. First, stability of these inequalities with respect to perturbation of the measure is obtained. The extent of the perturbation is measured using several different distances between perturbed and original measure, such as a onesided L ∞ bound on the ratio between their densities, Wasserstein distances, and KullbackLeibler divergence. In particular, an extension of the Holley–Stroock perturbation lemma for the logSobolev inequality is obtained. Second, the equivalence of TransportEntropy inequalities with different cost functions is verified, by obtaining a reverse Jensen type inequality. In view of a recent result of Gozlan, this is used to obtain tensorization properties of concentration inequalities with respect to various productmetrics, and the tensorization result for isoperimetric inequalities of Barthe–Cattiaux–Roberto is easily recovered. Some further applications are also described. The main tool used is a previous precise result on the equivalence between concentration and isoperimetric inequalities in the described setting.
Isoperimetry for spherically symmetric logconcave probability measures
, 2009
"... We prove an isoperimetric inequality for probability measures µ on Rn with density proportional to exp(−φ(λx)), where x  is the euclidean norm on Rn and φ is a nondecreasing convex function. It applies in particular when φ(x) = xα with α ≥ 1. Under mild assumptions on φ, the inequality is dime ..."
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Cited by 6 (0 self)
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We prove an isoperimetric inequality for probability measures µ on Rn with density proportional to exp(−φ(λx)), where x  is the euclidean norm on Rn and φ is a nondecreasing convex function. It applies in particular when φ(x) = xα with α ≥ 1. Under mild assumptions on φ, the inequality is dimensionfree if λ is chosen such that the covariance of µ is the identity. 1
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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Cited by 6 (0 self)
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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