Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5772320 | Journal of Functional Analysis | 2017 | 50 Pages |
Abstract
In this article, the bi-Lipschitz embeddability of the sequence of countably branching diamond graphs (DkÏ)kâN is investigated. In particular it is shown that for every ε>0 and kâN, DkÏ 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 (DkÏ)kâN 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.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Florent Baudier, Ryan Causey, Stephen Dilworth, Denka Kutzarova, Nirina L. Randrianarivony, Thomas Schlumprecht, Sheng Zhang,