A coding of bundle graphs and their embeddings into Banach spaces

TL;DR

This paper characterizes key Banach space properties like superreflexivity and asymptotic uniform convexifiability through the embeddability of bundle graphs, providing new insights into their geometric structure and embedding distortions.

Contribution

It introduces a novel graph-based framework to characterize Banach space properties via embeddings of bundle graphs, extending known results and providing explicit distortion bounds.

Findings

Superreflexivity characterized by non-embeddability of bundle graphs.

Asymptotic uniform convexifiability characterized within reflexive spaces.

Countably-branching bundle graphs embed into L1 with distortion ≤ 2.

Abstract

The purpose of this article is to generalize some known characterizations of Banach space properties in terms of graph preclusion. In particular, it is shown that superreflexivity can be characterized by the non-equi-bi-Lipschitz embeddability of any family of bundle graphs generated by a nontrivial finitely-branching bundle graph. It is likewise shown that asymptotic uniform convexifiability can be characterized within the class of reflexive Banach spaces with an unconditional asymptotic structure by the non-equi-bi-Lipschitz embeddability of any family of bundle graphs generated by a nontrivial $\aleph_0$-branching bundle graph. The best known distortions are recovered. For the specific case of $L_1$, it is shown that every countably-branching bundle graph bi-Lipschitzly embeds into $L_1$ with distortion no worse than $2$.