Results 1  10
of
12
Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry
, 2004
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Isoperimetry between exponential and Gaussian
 Electronic J. Prob
"... 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. 1 ..."
Abstract

Cited by 16 (7 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. 1
Spectral Gap, Logarithmic Sobolev Constant, and Geometric Bunds
 Surveys in Diff. Geom., Vol. IX, 219–240, Int
, 2004
"... We survey recent works on the connection between spectral gap and logarithmic Sobolev constants, and exponential integrability of Lipschitz functions. In particular, tools from measure concentration are used to describe bounds on the diameter of a (compact) Riemannian manifold and of Markov ch ..."
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Cited by 14 (0 self)
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We survey recent works on the connection between spectral gap and logarithmic Sobolev constants, and exponential integrability of Lipschitz functions. In particular, tools from measure concentration are used to describe bounds on the diameter of a (compact) Riemannian manifold and of Markov chains in terms of the first eigenvalue of the Laplacian and the logarithmic Sobolev constant. We examine similarly dimension free isoperimetric bounds using these parameters.
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 12 (3 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
H 1 AND BMO FOR CERTAIN LOCALLY DOUBLING METRIC MEASURE SPACES OF FINITE MEASURE
, 811
"... Abstract. In a previous paper the authors developed a H 1 −BMO theory for unbounded metric measure spaces (M, ρ, µ) of infinite measure that are locally doubling and satisfy two geometric properties, called “approximate midpoint” property and “isoperimetric ” property. In this paper we develop a sim ..."
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Cited by 2 (2 self)
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Abstract. In a previous paper the authors developed a H 1 −BMO theory for unbounded metric measure spaces (M, ρ, µ) of infinite measure that are locally doubling and satisfy two geometric properties, called “approximate midpoint” property and “isoperimetric ” property. In this paper we develop a similar theory for spaces of finite measure. We prove that all the results that hold in the infinite measure case have their counterparts in the finite measure case. Finally, we show that the theory applies to a class of unbounded, complete Riemannian manifolds of finite measure and to a class of metric measure spaces of the form (R d, ρϕ, µϕ), where dµϕ = e −ϕ dx and ρϕ is the Riemannian metric corresponding to the length element ds 2 = (1+∇ϕ) 2 (dx 2 1 + · · ·+ dx2 d). This generalizes previous work of the last two authors for the Gauss space. 1.
Properties of Isoperimetric, Functional and TransportEntropy Inequalities Via Concentration
, 909
"... 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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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. 1
POINTWISE SYMMETRIZATION INEQUALITIES FOR SOBOLEV FUNCTIONS AND APPLICATIONS
, 908
"... Abstract. 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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Abstract. 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.
A Poincaré Inequality on Loop Spaces
, 2009
"... We investigate properties of measures in infinite dimensional spaces in terms of Poincaré inequalities. A Poincaré inequality states that the L 2 variance of an admissible function is controlled by the homogeneous H 1 norm. In the case of Loop spaces, it was observed by L. Gross [17] that the homoge ..."
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We investigate properties of measures in infinite dimensional spaces in terms of Poincaré inequalities. A Poincaré inequality states that the L 2 variance of an admissible function is controlled by the homogeneous H 1 norm. In the case of Loop spaces, it was observed by L. Gross [17] that the homogeneous H 1 norm alone may not control the L 2 norm and a potential term involving the end value of the Brownian bridge is introduced. Aida, on the other hand, introduced a weight on the Dirichlet form. We show that Aida’s modified Logarithmic Sobolev inequality implies weak Logarithmic Sobolev Inequalities and weak Poincaré inequalities with precise estimates on the order of convergence. The order of convergence in the weak Sobolev inequalities are related to weak L 1 estimates on the weight function. This and a relation between Logarithmic Sobolev inequalities and weak Poincaré inequalities lead to a Poincaré inequality on the loop space over certain manifolds.