We study some problems on exponential integrability, concentration of measure, and transportation cost related to logarithmic Sobolev inequalities. On the real line, we then give a characterization of those probablity measures which satisfy these inequalities.

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A dimension free lower bound is found for isoperimetric constants of product probability measures. From this, some analytic inequalities are derived. 1 Introduction Let (X; d) be a metric space equipped with a non--atomic, separable Borel probability measure ¯. In the present paper we study the quantity Is(¯) = inf ¯ + (A) min(¯(A); 1 \Gamma ¯(A)) (1:1) Key words: Isoperimetry, Poincar'e Inequalities, Cheeger's inequality, Khinchine-- Kahane inequality, Holder's Inequality y Research supported in part by the ISF grant NZX000 and NZX300. This author enjoyed the hospitality of the Faculty of Wiskunde and Informatica, Free University of Amsterdam, while part of this research was carried out. z Research supported in part by an NSF Postdoctoral Fellowship. This author enjoyed the hospitality of Le Cermics, ENPC, France, of the Steklov Mathematical Institute (Sankt Petersburg branch) and of the Department of Mathematics, University of Syktyvkar, Russia, while part of this research...

Abstract. We consider random tries and random patricia trees constructed from n independent strings of symbols drawn from any distribution on any discrete space. If Hn is the height of this tree, we show that Hn/E{Hn} tends to one in probability. Additional tail inequalities are given for the height, depth, size, and profile of these trees and ordinary tries that apply without any conditions on the string distributions—they need not even be identically distributed.

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

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

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

. We prove a concentration inequality for functions, Lipschitz with respect to the Euclidean metric, on the ball of ` n p , 1 p ! 2 equipped with the normalized Lebesgue measure. 1 Introduction In [SZ] the authors proved an inequality which can be interpreted as giving the right order of the tail distribution of the ` n q norm on the ` n p ball equipped with the normalized Lebesgue measure. More precisely and specializing to the case of q = 2 and p = 1, we proved there that (fx : kxk 2 ? tg) is bounded above by C exp(\Gammactn) for t ? T= p n (where C; c and T are absolute constants) and bounded from below by a similar quantity (with different absolute constants). The measure can be either the normalized Lebesgue measure on B n 1 - the ball of ` n 1 or the normalized Lebesgue measure on @B n 1 - the sphere of ` n 1 . (For other p's the relevant measure on the sphere is a different one than the usual surface measure.) Following a question of M. Gromov, we generalize her...

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

Introduction Motivation, Examples, Statements of Results It is well known that the Sobolev inequality Z R n jrf(x)jd¯(x) n! 1=n n `Z R n jf(x)j n n\Gamma1 d¯(x) 'n\Gamma1 n ; n 2; (1.1) where f is a compactly supported smooth function on the Euclidean space R n , where ¯ is the Lebesgue measure in R n , and where ! n is the volume of the unit ball in R n , is equivalent to the isoperimetric property of the balls in R n . This isoperimetric property