On the geometry of the countably branching diamond graphs

TL;DR

This paper characterizes the embeddability of countably branching diamond graphs into Banach spaces, revealing conditions under which these graphs can or cannot be embedded bi-Lipschitzly, with implications for Banach space geometry.

Contribution

It provides a metric characterization of asymptotically uniformly convexifiable spaces via graph preclusion, advancing understanding of bi-Lipschitz embeddability in Banach space theory.

Findings

Countably branching diamond graphs embed bi-Lipschitzly into certain Banach spaces with controlled distortion.

These graphs do not embed into spaces with asymptotically midpoint uniformly convex norms.

Results have applications to embeddings into $L_p$-spaces and renorming problems.

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.