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.
Subject
Banachnonlinear
metric
coarse
uniform
Lipschitz
embedding
paracompact
superstable
bundle
graph
Citation
Swift, Andrew Thomas (2018). On Some Problems in the Nonlinear Geometry of Banach Spaces. Doctoral dissertation, Texas A & M University. Available electronically from https : / /hdl .handle .net /1969 .1 /173913.