Asymptotic geometry of Banach spaces and uniform quotient maps

Asymptotic Geometry of Banach Spaces and Uniform Quotient Maps

Recently, Lima and Randrianarivony pointed out the role of the property (β) of Rolewicz in nonlinear quotient problems, and answered a ten-year-old question of Bates, Johnson, Lindenstrauss, Preiss and Schechtman. In the present paper, we prove that the modulus of asymptotic uniform smoothness of the range space of a uniform quotient map can be compared with the modulus of (β) of the domain spa...

Uniform Embeddings, Homeomorphisms and Quotient Maps between Banach Spaces (a Short Survey)

Uniform Homeomorphisms of Banach Spaces and Asymptotic Structure

We give a general result on the behavior of spreading models in Banach spaces which coarse Lipschitz-embed into asymptotically uniformly convex spaces. We use this result to study the uniqueness of the uniform structure in p-sums of finite-dimensional spaces for 1 < p < ∞; in particular we give some new examples of spaces with unique uniform structure.

Asymptotic isoperimetry on groups and uniform embeddings into Banach spaces

We characterize the asymptotic behaviour of the compression associated to a uniform embedding into some Lp-space for a large class of groups including connected Lie groups with exponential growth and wordhyperbolic finitely generated groups. This also provides new and optimal estimates for the compression of a uniform embedding of the infinite 3regular tree into some Lp-space. The main part of ...

Randomized Series and Geometry of Banach Spaces

Abstract. We study some properties of the randomized series and their applications to the geometric structure of Banach spaces. For n ≥ 2 and 1 < p < ∞, it is shown that l ∞ is representable in a Banach space X if and only if it is representable in the Lebesgue-Bochner Lp(X). New criteria for various convexity properties in Banach spaces are also studied. It is proved that a Banach lattice E is...