Contents 1. Introduction 2 2. Metric type 7 2.1. Type for metric spaces 8 2.2. The geometric puzzle 11 2.3. Pisier's argument 12 2.4. Unique obstructions to type 17 3. Metric cotype 19 4. Markov type and cotype 22 4.1. Lipschitz extension via Markov type and cotype 27 5. Markov convexity 29 6. Metric smoothness? 32 7. Bourgain's discretization problem 32 8. Nonlinear Dvoretzky theorems 33 8.1. ...

In this paper, we introduce a new geometric property (UÃ2) and we show that if a separable Banach space has this property, then both X and its dual X∗ have the weak fixed point property. We also prove that a uniformly Gateaux differentiable Banach space has property (UÃ2) and that if X∗ has property (UÃ2), then X has the (UKK)-property. Criteria for Orlicz spaces to have the properties (UA2), (...

For rather general moduli of smoothness ρ, like ρ(h)=h ln(c/h), we consider the Hölder spaces Hρ(B) of functions [0, 1] → B where B is a separable Banach space. We establish an isomorphism between Hρ(B) and some sequence Banach space. With this analytical tool, we follow a very natural way to study, in terms of second differences, the existence of a version in Hρ(B) for a given random field. 20...

We consider metric measure spaces satisfing a doubling condition and a Poincaré inequality in the upper gradient sense. We show that the results of [Che99] on differentiability of real valued Lipschitz functions and the resulting bi-Lipschitz nonembedding theorems for finite dimensional vector space targets extend to Banach space targets having what we term a good finite dimensional approximati...

We explore the existence of uniformly continuous sections for quotient maps. Using this approach we are able to give a number of new examples in the theory of the uniform structure of Banach spaces. We show for example that there are two non-isomorphic separable L1-subspaces of 1 which are uniformly homeomorphic. We also prove the existence of two coarsely homeomorphic Banach spaces (i.e. with ...

The notions of relaxed submonotone and relaxed monotone mappings in Banach spaces are introduced and many of their properties are investigated. For example, the Clarke subdifferential of a locally Lipschitz function in a separable Banach space is relaxed submonotone on a residual subset. For example, it is shown that this property need not be valid on the whole space. We prove, under certain hy...

This paper shows necessary and sufficient conditions on subsets of variable exponent spaces Lp(?)(?) in order to be weakly compact. Useful criteria are given extending Andô results for Orlicz spaces. As application, we prove that all separable Banach-Saks. Also, L-weakly compact inclusions between studied.

In this paper we extend a result of Šemrl stating that every 2-local automorphism of the full operator algebra on a separable infinite dimensional Hilbert space is an automorphism. In fact, besides separable Hilbert spaces, we obtain the same conclusion for the much larger class of Banach spaces with Schauder bases. The proof rests on an analogous statement concerning the 2-local automorphisms ...