2017
DOI: 10.1016/j.jfa.2017.05.013
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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… Show more

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Cited by 10 publications
(28 citation statements)
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References 44 publications
(91 reference statements)
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“…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.…”
Section: Introductionmentioning
confidence: 83%
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“…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.…”
Section: Introductionmentioning
confidence: 83%
“…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.…”
Section: Notation and Definitionsmentioning
confidence: 99%
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“…Also, in recent years diamond graphs of high branching have appeared naturally in different contexts, cf. [4,26,43].…”
Section: Introductionmentioning
confidence: 99%
“…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.…”
Section: Final Remarks and Open Problemsmentioning
confidence: 99%