Properties of Isoperimetric, Functional and Transport-Entropy Inequalities Via Concentration (2009)
Various properties of isoperimetric, functional, Transport-Entropy and concentration inequalities are studied on a Riemannian manifold equipped with a measure, whose generalized Ricci curvature is...
A converse to Maz'ya's inequality for capacities under curvature lower bound (2009)
We survey some classical inequalities due to Maz'ya relating isocapacitary inequalities with their functional and isoperimetric counterparts in a measure-metric space setting, and extend Maz'ya's...
It is well known that isoperimetric inequalities imply in a very general measure-metric-space setting appropriate concentration inequalities. The former bound the boundary measure of sets as a...
On the role of convexity in functional and isoperimetric inequalities (2009)
This is a continuation of our previous work [Preprint, 2008, http://arxiv.org/abs/0712.4092]. It is well known that various isoperimetric inequalities imply their functional ‘counterparts’, but...
On the role of Convexity in Functional and Isoperimetric Inequalities (2008)
This is a continuation of our previous work 0712.4092. It is well known that various isoperimetric inequalities imply their functional ``counterparts'', but in general this is not an equivalence. We...
On the role of Convexity in Isoperimetry, Spectral-Gap and Concentration (2007)
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...
An isoperimetric inequality for uniformly log-concave measures and uniformly convex bodies (2007)
We prove an isoperimetric inequality for the uniform measure on a uniformly convex body and for a class of uniformly log-concave measures (that we introduce). These inequalities imply (up to...
Generalized Intersection Bodies are not Equivalent (2007)
In 2000, A. Koldobsky asked whether two types of generalizations of the notion of an intersection-body, are in fact equivalent. The structures of these two types of generalized intersection-bodies...
On volume distribution in 2-convex bodies (2006)
Klartag, Boaz, Milman, Emanuel
We consider convex sets whose modulus of convexity is uniformly quadratic. First, we observe several interesting relations between different positions of such ``2-convex'' bodies; in particular, the...
On Gaussian Marginals of Uniformly Convex Bodies (2006)
Recently, Bo'az Klartag showed that arbitrary convex bodies have Gaussian marginals in most directions. We show that Klartag's quantitative estimates may be improved for many uniformly convex bodies....
A remark on two duality relations (2006)
We remark that an easy combination of two known results yields a positive answer, up to log(n) terms, to a duality conjecture that goes back to Pietsch. In particular, we show that for any two...
Dual Mixed Volumes and the Slicing Problem (2005)
We develop a technique using dual mixed-volumes to study the isotropic constants of some classes of spaces. In particular, we recover, strengthen and generalize results of Ball and Junge concerning...
A comment on the low-dimensional Busemann-Petty problem (2005)
The generalized Busemann-Petty problem asks whether centrally-symmetric convex bodies having larger volume of all m-dimensional sections necessarily have larger volume. When m>3 this is known to be...
Generalized Intersection Bodies (2005)
We study the structures of two types of generalizations of intersection-bodies and the problem of whether they are in fact equivalent. Intersection-bodies were introduced by Lutwak and played a key...