Results 1  10
of
17
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 ..."
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Cited by 15 (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
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 14 (4 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
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 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 4 (3 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
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.
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 1 (0 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.
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.