On Some Problems in the Nonlinear Geometry of Banach Spaces

Abstract

Two general problems in the nonlinear geometry of Banach spaces are to determine the relationship between uniform and coarse embeddings and to characterize local/asymptotic properties in terms of metric structure. The purpose of this research is to investigate these problems and to contribute to a better overall understanding of the structure of Banach spaces and metric spaces. First, we investigate the relationship between the small-scale and large-scale structures of Cv0(k). In 1994, Jan Pelant proved that a metric property related to the notion of paracompactness called the uniform Stone property characterizes a metric space’s uniform embeddability into Cv0(k) for some cardinality k. We show that coarse Lipschitz embeddability of a metric space into Cv(0k) can be characterized in a similar manner. We also show that coarse, uniform, and bi-Lipschitz embeddability into Cv0(k) are equivalent notions for normed linear spaces. Next, we investigate the relationship between the small-scale and large-scale structures of superstable Banach spaces. In 1983, Yves Raynaud showed that if a Banach space uniformly embeds into a superstable Banach space, then X must contain an isomorphic copy of ℒvp, for some p € [1, ∞). Using similar methods, we show that if a Banach space coarsely embeds into a superstable Banach space, then X has a spreading model isomorphic to ℒvp, for some p € [1, ∞). This implies the existence of reflexive Banach spaces that do not coarsely embed into any superstable Banach space. Lastly, we define a class of graphs, which we call the “bundle graphs”, and use this to generalize some known metric characterizations of Banach space properties in terms of graph preclusion. In particular, we generalize the characterizations of superreflexivity within the class of Banach spaces and asymptotic uniform convexifiability within the class of reflexive Banach spaces with unconditional asymptotic structure. For the specific case of Lv1, we show that every Nv0-branching bundle graph bi-Lipschitzly embeds into Lv1 with distortion no worse than 2.