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- Emanuel Milman
- 2008

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… (More)

- Emanuel Milman
- 2008

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 role in the solution of the Busemann-Petty problem. A natural geometric generalization of this problem considered by Zhang, led him to introduce one type of… (More)

We prove an isoperimetric inequality for uniformly log-concave measures and for the uniform measure on a uniformly convex body. These inequalities imply the log-Sobolev inequalities proved by Bobkov and Ledoux [12] and Bobkov and Zegarlinski [13]. We also recover a concentration inequality for uniformly convex bodies, similar to that proved by Gromov and… (More)

- Emanuel Milman
- 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 log-concave probability measures in Euclidean space), the latter implication can in fact be reversed for… (More)

- EMANUEL MILMAN
- 2006

We show that many uniformly convex bodies have Gaussian marginals in most directions in a strong sense, which takes into account the tails of the distributions. These include uniformly convex bodies with power type 2, and power type p > 2 with some additional type condition. In particular, all unit-balls of subspaces of L p for 1 < p < ∞ have Gaussian… (More)

- EMANUEL MILMAN
- 2007

In [Kol00], 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 have been studied in [Mil06b], providing substantial evidence for a positive answer to this question. The purpose of this note is to construct a counterexample… (More)

- EMANUEL MILMAN
- 2008

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 the isotropic constants of subspaces and quotients of L p and related spaces. An extension of these results to negative values of p is also obtained, using… (More)

- BO’AZ KLARTAG, EMANUEL MILMAN
- 2006

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 isotropic position is a finite volume-ratio position for these bodies. Second, we prove that high dimensional 2-convex bodies posses one-dimensional marginals… (More)