Results 1  10
of
13
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 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
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
An isoperimetric inequality for uniformly logconcave measures and uniformly convex bodies
, 2008
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On the isoperimetric problem in Euclidean space with density Calc
 Var. Partial Differential Equations
, 2008
"... ABSTRACT. We study the isoperimetric problem for Euclidean space endowed with a continuous density. In dimension one, we characterize isoperimetric regions for a unimodal density. In higher dimensions, we prove existence results and we derive stability conditions, which lead to the conjecture that f ..."
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Cited by 10 (1 self)
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ABSTRACT. We study the isoperimetric problem for Euclidean space endowed with a continuous density. In dimension one, we characterize isoperimetric regions for a unimodal density. In higher dimensions, we prove existence results and we derive stability conditions, which lead to the conjecture that for a radial logconvex density, balls about the origin are isoperimetric regions. Finally, we prove this conjecture and the uniqueness of minimizers for the density exp(x  2) by using symmetrization techniques. 1.
Extremal properties of central halfspaces for product measures
 J. Funct. Anal
, 2001
"... Extremal properties of central halfspaces for product measures F. Barthe\Lambda Abstract We deal with the isoperimetric and the shift problem for subsets of measure one half in product probability spaces. We prove that the canonical central halfspaces are extremal in particular cases: products of ..."
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Cited by 7 (2 self)
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Extremal properties of central halfspaces for product measures F. Barthe\Lambda Abstract We deal with the isoperimetric and the shift problem for subsets of measure one half in product probability spaces. We prove that the canonical central halfspaces are extremal in particular cases: products of logconcave measures on the real line satisfying precise conditions and products of uniform measures on spheres, or balls. As a corollary, we improve the known logSobolev constants for Euclidean balls. We also give some new results about the related question of estimating the volume of sections of unit balls of `psums of Minkowski spaces.
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 7 (1 self)
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Large deviations and isoperimetric inequalities are considered for probability distributions,
satisfying convexity conditions of the BrunnMinkowskitype.
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 4 (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
GEOMETRIC INFLUENCES
"... Abstract. We present a new definition of influences in product spaces of continuous distributions. Our definition is geometric, and for monotone sets it is identical with the measure of the boundary with respect to uniform enlargement. We prove analogues of the KahnKalaiLinial (KKL) and Talagrand’ ..."
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Cited by 2 (1 self)
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Abstract. We present a new definition of influences in product spaces of continuous distributions. Our definition is geometric, and for monotone sets it is identical with the measure of the boundary with respect to uniform enlargement. We prove analogues of the KahnKalaiLinial (KKL) and Talagrand’s influence sum bounds for the new definition. We further prove an analogue of a result of Friedgut showing that sets with small “influence sum ” are essentially determined by a small number of coordinates. In particular, we establish the following tight analogue of the KKL bound: for any set in R n of Gaussian measure t, there exists a coordinate i such that the ith geometric influence of the set is at least ct(1−t) √ log n/n, where c is a universal constant. This result is then used to obtain an isoperimetric inequality for the Gaussian measure on R n and the class of sets invariant under transitive permutation group of the coordinates. 1.
Dimension Free Weak Concentration Of Measure Phenomenon
, 1995
"... For product probability measures ¯ n , we obtain necessary and sufficient conditions (in terms of ¯) for dimension free isoperimetric inequalities of the form ¯ n (A + h[\Gamma1; 1] n ) R h (¯ n (A)) to hold; for a function R such that R(p) ? p, for all (some) p 2 (0; 1), and for h ? 0 lar ..."
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For product probability measures ¯ n , we obtain necessary and sufficient conditions (in terms of ¯) for dimension free isoperimetric inequalities of the form ¯ n (A + h[\Gamma1; 1] n ) R h (¯ n (A)) to hold; for a function R such that R(p) ? p, for all (some) p 2 (0; 1), and for h ? 0 large enough. Some questions related to the concentration of measure phenomenon are also discussed. 1 Introduction Let ¯ be a probability measure on the real line R, and let ¯ n be the nfold tensor product of ¯ with itself. Given a notion of enlargement enl(A) for sets A, inequalities of isoperimetric type have the form ¯ n (enl(A)) R (n) (¯(A)): Moreover, if R = R (n) is dimension free, such inequalities are often viewed as concentration inequalities. One question of interest which will be addressed here is, whether or not, there exists such a function R (of course, such that R(p) ? p). Besides the measure, the answer essentially depends on the enlargement. Usually, it is built wi...