Results 1  10
of
20
Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry
, 2004
"... ..."
(Show Context)
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 ..."
Abstract

Cited by 19 (5 self)
 Add to MetaCart
(Show Context)
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
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. ..."
Abstract

Cited by 17 (8 self)
 Add to MetaCart
(Show Context)
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
"... ..."
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 ..."
Abstract

Cited by 13 (1 self)
 Add to MetaCart
(Show Context)
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.
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. ..."
Abstract

Cited by 11 (1 self)
 Add to MetaCart
Large deviations and isoperimetric inequalities are considered for probability distributions,
satisfying convexity conditions of the BrunnMinkowskitype.
Extremal properties of central halfspaces for product measures
 J. Funct. Anal
"... Extremal properties of central halfspaces for product measures ..."
Abstract

Cited by 9 (2 self)
 Add to MetaCart
(Show Context)
Extremal properties of central halfspaces for product measures
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) ..."
Abstract

Cited by 8 (4 self)
 Add to MetaCart
(Show Context)
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
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 ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
(Show Context)
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
Isoperimetric and Concentration Inequalities Part I: Equivalence under Curvature Lower Bound
, 2009
"... It is well known that isoperimetric inequalities imply in a very general measuremetricspace setting appropriate concentration inequalities. The former bound the boundary measure of sets as a function of their measure, whereas the latter bound the measure of sets separated from sets having half the ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
(Show Context)
It is well known that isoperimetric inequalities imply in a very general measuremetricspace setting appropriate concentration inequalities. The former bound the boundary measure of sets as a function of their measure, whereas the latter bound the measure of sets separated from sets having half the total measure, as a function of their mutual distance. We show that under a lower bound condition on the Bakry – Émery curvature tensor of a Riemannian manifold equipped with a density, completely general concentration inequalities imply back their isoperimetric counterparts, up to dimension independent bounds. As a corollary, we can recover and extend all previously known (dimension dependent) results by generalizing an isoperimetric inequality of Bobkov, and provide a new proof that under natural convexity assumptions, arbitrarily weak concentration implies a dimension independent linear isoperimetric inequality. Further applications will be described in a subsequent work. Contrary to previous attempts in this direction, our method is entirely geometric, continuing the approach set forth by Gromov and adapted to the manifoldwithdensity setting by Morgan. 1