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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 half-spaces are approximate solutions of the isoperimetric problem. 1

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 log-convex 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.

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Extremal properties of central half-spaces 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 half-spaces are extremal in particular cases: products of log-concave measures on the real line satisfying precise conditions and products of uniform measures on spheres, or balls. As a corollary, we improve the known log-Sobolev constants for Euclidean balls. We also give some new results about the related question of estimating the volume of sections of unit balls of `p-sums of Minkowski spaces.

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 a-priori weakest requirement that Lipschitz functions have arbitrarily slow uniform tail-decay, 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 “on-average ” 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, ∞) curvature-dimension condition of Bakry-Émery. 1

Large deviations and isoperimetric inequalities are considered for probability distributions, satisfying convexity conditions of the Brunn-Minkowski-type.

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

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 n--fold 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...

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 Kahn-Kalai-Linial (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 i-th 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.