We construct a locally finite graph and a bounded geometry metric space which do not admit a quasi-isometric embedding into any uniformly convex Banach space. Connections with the geometry of c0 and superreflexivity are discussed. The question of coarse embeddability into uniformly convex Banach spaces became interesting after the recent work of G. Kasparov and G. Yu, who showed the coarse Novi...

Abstract In hyperbolic geometry there are several concepts to measure the breadth or width of a convex set. first part paper we collect them and compare their properties. Than introduce new concept thickness body. Correspondingly, define three classes bodies, bodies constant with, diameter having shadow property, respectively. We prove that property follows fulfilment both stronger as width. la...

Abstract We introduce the concept of $\epsilon $-uncontrollability for random linear systems, i.e. systems in which usual matrices have been replaced by matrices. also estimate $\varepsilon case where come from Gaussian orthogonal ensemble. Our proof utilizes tools theory, probability theory and convex geometry.

If M is a closed Nil geometry 3-manifold then π 1 (M) is almost convex with respect to a fairly simple " geometric " generating set. If G is a central extension or a Z extension of a word hyperbolic group, then G is also almost convex with respect to some generating set. Combining these with previously known results shows that if M is a closed 3-manifold with one of Thurston's eight geometries,...

Using notions of Minkowski geometry (i.e., of the geometry of finite dimensional Banach spaces) we find new characterizations of centrally symmetric convex bodies, equiframed curves, bodies of constant width and certain convex bodies with modified constant width property. In particular, we show that straightforward extensions of some properties of bodies of constant Euclidean width are also val...

In the development of the subject of Geometric Probabilities, there was always a close relationship to Convex Geometry. In these lectures, I want to demonstrate this relationship with a number of examples. These examples are of different types, since I want to cover various aspects. I start with hitting probabilities for convex bodies, which can be treated by means of integral geometry. Then I ...

Finding the nearest point on a given 2D or 3D object computation is a problem that arises in many computer graphics applications, such as interactive modeling, rendering of implicit surfaces or collision detection. Unfortunately, for objects with complex geometry such as curves or surfaces, the computation cost becomes prohibitive as a minimization process acting in multiple dimensions has to b...