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 ..."
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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
Concentration for independent random variables with heavy tails
 AMRX
, 2005
"... If a random variable is not exponentially integrable, it is known that no concentration inequality holds for an infinite sequence of independent copies. Under mild conditions, we establish concentration inequalities for finite sequences of n independent copies, with good dependence in n. 1 ..."
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Cited by 14 (8 self)
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If a random variable is not exponentially integrable, it is known that no concentration inequality holds for an infinite sequence of independent copies. Under mild conditions, we establish concentration inequalities for finite sequences of n independent copies, with good dependence in n. 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
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.
On the infimum convolution inequality
"... In the paper we study the infimum convolution inequalites. Such an inequality was first introduced by B. Maurey to give the optimal concentration of measure behaviour for the product exponential measure. We show how IC inequalities are tied to concentration and study the optimal cost functions for a ..."
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Cited by 7 (0 self)
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In the paper we study the infimum convolution inequalites. Such an inequality was first introduced by B. Maurey to give the optimal concentration of measure behaviour for the product exponential measure. We show how IC inequalities are tied to concentration and study the optimal cost functions for an arbitrary probability measure µ. In particular, we show the optimal IC inequality for product log–concave measures and for uniform measures on the ℓ n p balls. Such an optimal inequality implies, for a given measure, in particular the Central Limit Theorem of Klartag and the tail estimates of Paouris. 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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Cited by 5 (4 self)
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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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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
SECONDORDER DERIVATIVES AND REARRANGEMENTS
 VOL. 105, NO. 3 DUKE MATHEMATICAL JOURNAL
, 2000
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