The nonlinear geometry of Banach spaces
This highly informative survey paper discusses some recent developments in the nonlinear theory of Banach spaces. The basic problem here is to recover the linear structure of a Banach space from properties of the underlying metric space. The author presents various (mostly rather deep) results in this area due to J.~Lindenstrauss, P.~Enflo, J.~Bourgain, W.B.~Johnson, G.~Schechtman, A.~Naor and himself (and, needless to say, many others). In detail, the paper discusses the following topics. Section~1 collects some background material. Section~2 contains results connected with the problem of whether two Banach spaces are linearly isomorphic if they are Lipschitz (or merely uniformly) homeomorphic. Here, two approaches are introduced: on the one hand, methods relying on differentiability properties of Lipschitz maps, on the other hand, the approach through the Arens-Eells space (also known as Lipschitz free space). The theme of Section~3 is to investigate nonlinear analogues of classical Banach space notions (such as nonlinear type and cotype), absolute Lipschitz retracts, and nonlinear versions of Maurey's extension theorem. Finally, Section~4 is devoted to uniform and coarse embeddings of the unit ball and indeed general metric spaces. Various problems are mentioned throughout the text. The paper is very well written and highly recommended to everyone who wishes to study this new area of Banach space theory.
nonlinear classification of Banach spaces
scientific article; zbMATH DE number 5289678