We prove that if a metric probability space with a usual concentration property embeds into a finite dimensional Banach space X , then X has a Euclidean subspace of a proportional dimension. In particular this yields a new characterization of weak cotype 2. We also find optimal lower estimates on embeddings of metric spaces with concentration properties into l ∞, generalizing estimates of Bourg...

In this paperwe prove that the /~.,-cube can be (1 + s)-embedded into any 1 -subsyntmetrie C(s>n.dimensional normed space. Marcus and Pisier in [5]iniciated tite study of tite geometry ob finite metric spaces. Bourgain, Milman and Wolbson introduced a new notion of metnc type and developed tite non-linear titeory of Banacit spaces (see [2]and [7]). AII titese themes have been studied more inten...

In a partially ordered semigroup with the duality (or polarity) transform, it is possible to define a generalisation of continued fractions. General sufficient conditions for convergence of continued fractions are provided. Two particular applications concern the cases of convex sets with the Minkowski addition and the polarity transform and the family of non-negative convex functions with the ...

We present some sufficient conditions for which a Banach space X has normal structure in terms of the modulus of U-convexity, modulus of W∗-convexity, and the coefficient R 1, X , which generalized some well-known results. Furthermore the relationship between modulus of convexity, modulus of smoothness, and Gao’s constant is considered, meanwhile the exact value of Milman modulus has been obtai...

A game on a convex geometry is a real-valued function de®ned on the family L of the closed sets of a closure operator which satis®es the ®nite Minkowski±Krein±Milman property. If L is the Boolean algebra 2 N then we obtain a n-person cooperative game. We will introduce convex and quasi-convex games on convex geometries and we will study some properties of the core and the Weber set of these games.

We characterize the duality of convex bodies in d-dimensional Euclidean vector space, viewed as a mapping from the space of convex bodies containing the origin in the interior into the same space. The question for such a characterization was posed by Vitali Milman. Sufficient for a characterization, up to a trivial exception and the composition with a linear transformation, is the property that...

A generalization of Lozanovskii's result is proved. Let E be k-dimensional sub-space of an n-dimensional Banach space with unconditional basis. 1 k ≤ e n k 2. This answers a question of V. Milman which appeared during a GAFA seminar talk about the hyperplane problem. We add logarithmical estimates concerning the hyperplane conjecture for proportional subspaces and quotients of Banach spaces wit...

Doubly stochastic measures are Borel probability measures on the unit square which push forward via the canonical projections to Lebesgue measure on each axis. The set of doubly stochastic measures is convex, so its extreme points are of particular interest. I review necessary and sufficient conditions for a set to support an extremal doubly stochastic measure, and include a proof that such a s...