On the geometry of the countably branching diamond graphs

Abstract

In this article, the bi-Lipschitz embeddability of the sequence of countably branching diamond graphs $(D_k^\omega)_{k\in\mathbb{N}}$ is investigated. In particular it is shown that for every $\varepsilon>0$ and $k\in\mathbb{N}$, $D_k^\omega$ embeds bi-Lipschiztly with distortion at most $6(1+\varepsilon)$ into any reflexive Banach space with an unconditional asymptotic structure that does not admit an equivalent asymptotically uniformly convex norm. On the other hand it is shown that the sequence $(D_k^\omega)_{k\in\mathbb{N}}$ does not admit an equi-bi-Lipschitz embedding into any Banach space that has an equivalent asymptotically midpoint uniformly convex norm. Combining these two results one obtains a metric characterization in terms of graph preclusion of the class of asymptotically uniformly convexifiable spaces, within the class of separable reflexive Banach spaces with an unconditional asymptotic structure. Applications to bi-Lipschitz embeddability into $L_p$-spaces and to some problems in renorming theory are also discussed.Comment: 44 pages, 3 tables, 2 figure

A Coding of Bundle Graphs and Their Embeddings Into Banach Spaces

“…We'll then derive a formula for the graph metric in terms of this labelling. In Sections 3 and 4, we'll generalize two results in [3]. In Section 3 we'll show that every countably-branching bundle graph is bi-Lipschitzly embeddable into any Banach space with a good ℓ ∞ -tree with distortion bounded above by a constant depending only on the good ℓ ∞ -tree, which implies a more general characterization of asymptotic uniform convexifiability for the class of reflexive Banach spaces with an unconditional asymptotic structure.…”

“…We'll give a proof of this fact in Section 6. Thus the diamond and Laakso graphs used in [7], [9], and [3] are all examples of bundle graphs.…”

“…Since then, the non-equi-bi-Lipschitz embeddability of several other families of graphs have also been shown to characterize superreflexivity ( [2], [7], [9]). In [3], F. Baudier et al proved that the non-equi-bi-Lipschitz embeddability of the family of ℵ 0 -branching diamond graphs characterizes the asymptotic uniform convexifiability of refexive Banach spaces with an unconditional asymptotic structure. They also show that this same family of graphs is equi-bi-Lipschitzly embeddable into L 1 .…”

A new approach to low-distortion embeddings of finite metric spaces into non-superreflexive Banach spaces

“…Also, in recent years diamond graphs of high branching have appeared naturally in different contexts, cf. [4,26,43].…”

A New Coarsely Rigid Class of Banach Spaces

“…Recall that, as it was shown in [4], within the class of reflexive Banach spaces the subclass of reflexive spaces that admit an equivalent asymptotic uniformly smooth norm (i.e., they are AUS-able) and admit an equivalent asymptotic uniformly convex norm (i.e., they are AUC-able) is coarse Lipschitzly rigid. It was later proved in [3] that, within the class of reflexive spaces with an unconditional asymptotic structure, the subclass of such spaces that are additionally AUC-able is coarse Lipschitzly rigid. Whithin this context we are also inclined to study the metric properties of AUS-able spaces.…”