Related papers: On Matou\v{s}ek-like Embedding Obstructions of Cou…

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 $k\in\mathbb{N}$,…

Diamond graphs and binary trees are important examples in the theory of metric embeddings and also in the theory of metric characterizations of Banach spaces. Some results for these families of graphs are parallel to each other, for example…

A distortion lower bound of $\Omega(\log(h)^{1/p})$ is proven for embedding the complete countably branching hyperbolic tree of height $h$ into a Banach space admitting an equivalent norm satisfying property $(\beta)$ of Rolewicz with…

We introduce two ordinal indices that are linear invariants for Banach spaces: the dyadic tree index and the sprawling tree index. We show that they are also bi-Lipschitz invariants. In fact, we characterize their values in terms of…

The paper contains the following results and observations: (1) There exists a sequence of unweighted graphs $\{G_n\}_n$ with maximum degree 3 such that a Banach space $X$ has no nontrivial cotype iff $\{G_n\}_n$ admit uniformly bilipschitz…

In this note we extend to the quasi-reflexive setting the result of F. Baudier, N. Kalton and G. Lancien concerning the non-embeddability of the family of countably branching trees into reflexive Banach spaces whose Szlenk index and Szlenk…

In this note the result by A. Swift concerning the embeddability of countably branching bundle graphs into Banach spaces is extended from the context of reflexive spaces with an unconditional asymptotic structure to the context of dual…

An infinite dimensional notion of asymptotic structure is considered. This notion is developed in terms of trees and branches on Banach spaces. Every countably infinite countably branching tree $\mathcal T$ of a certain type on a space X is…

In this note, we study the geometry of the unit ball of the Banach space generated by the adequate family of all subsets of branches of the infinite binary tree, and answer several open questions related to slicely countably determined…

We show a distortion lower bound of $\Omega(\log(h)^{1/p})$ when embedding the countably branching hyperbolic tree of height $h$ into a Banach space with an equivalent norm satisfying Rolewicz property $(\beta)$ with modulus of power type…

For a fixed $K\gg 1$ and $n\in\mathbb{N}$, $n\gg 1$, we study metric spaces which admit embeddings with distortion $\le K$ into each $n$-dimensional Banach space. Classical examples include spaces embeddable into $\log n$-dimensional…

We find a class of metric structures which do not admit bilipschitz embeddings into Banach spaces with the Radon-Nikod\'ym property. Our proof relies on Chatterji's (1968) martingale characterization of the RNP and does not use the…

In 2021 Baudier, Motakis, Schlumprecht, and Zs\'ak proved that if a sequence of graphs $(G_k)_{k\in{\mathbb{N}}}$ contains the sequence of complete graphs with uniformly bounded distortion, then the sequence of lamplighter graphs on $G_k$'s…

The classical Banach--Mazur theorem asserts that every separable Banach space admits an isometric embedding into $C[0,1]$. It is also well known that every separable Banach space embeds isometrically into $\ell^\infty$. We show that such an…

The main purpose of the paper is to prove the following results: Let $A$ be a locally finite metric space whose finite subsets admit uniformly bilipschitz embeddings into a Banach space $X$. Then $A$ admits a bilipschitz embedding into $X$.…

For every $p\in(0,\infty)$, a new metric invariant called umbel $p$-convexity is introduced. The asymptotic notion of umbel convexity captures the geometry of countably branching trees, much in the same way as Markov convexity, the local…

Diamond graphs and Laakso graphs are important examples in the theory of metric embeddings. Many results for these families of graphs are similar to each other. In this connection, it is natural to ask whether one of these families admits…

We show the existence of Lipschitz-free spaces verifying the Point of Continuity Property with arbitrarily high weak-fragmentability index. For this purpose, we use a generalized construction of the countably branching diamond graphs. As a…

We combine Kirchheim's metric differentials with Cheeger charts in order to establish a non-embeddability principle for any collection $\mathcal C$ of Banach (or metric) spaces: if a metric measure space $X$ bi-Lipschitz embeds in some…

Let N and M be von Neumann algebras. It is proved that L^p(N) does not Banach embed in L^p(M) for N infinite, M finite, 1 < or = p < 2. The following considerably stronger result is obtained (which implies this, since the Schatten p-class…