On the geometry of the countably branching diamond graphs

Florent Baudier, Ryan Causey, Stephen Dilworth, Denka Kutzarova, Nirina L. Randrianarivony, Thomas Schlumprecht, Sheng Zhang

Research output: Contribution to journalArticlepeer-review


In this article, the bi-Lipschitz embeddability of the sequence of countably branching diamond graphs (Dk ω)k∈N is investigated. In particular it is shown that for every ε>0 and k∈N, Dk ω embeds bi-Lipschiztly with distortion at most 6(1+ε) 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 (Dk ω)k∈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 reflexive Banach spaces with an unconditional asymptotic structure. Applications to bi-Lipschitz embeddability into Lp-spaces and to some problems in renorming theory are also discussed.

Original languageEnglish (US)
Pages (from-to)3150-3199
Number of pages50
JournalJournal of Functional Analysis
Issue number10
StatePublished - Nov 15 2017


  • Asymptotic uniform convexity
  • Countably branching diamond graphs
  • Metric characterization
  • Unconditional asymptotic structure

ASJC Scopus subject areas

  • Analysis


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