Related papers: Metric embeddings of Laakso graphs into Banach spa…

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…

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…

We give sufficient conditions for a metric space to bilipschitz embed in L_1. In particular, if X is a length space and there is a Lipschitz map u:X--->R such that for every interval I in R, the connected components of the inverse image…

We show that any embedding of the level-k diamond graph of Newman and Rabinovich into $L_p$, $1 < p \le 2$, requires distortion at least $\sqrt{k(p-1) + 1}$. An immediate consequence is that there exist arbitrarily large n-point sets $X…

We characterize non-reflexive Banach spaces by a low-distortion (resp. isometric) embeddability of a certain metric graph up to a renorming. Also we study non-linear sufficient conditions for $\ell_1^n$ being $(1+\varepsilon)$-isomorphic to…

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 observe that embeddings into random metrics can be fruitfully used to study the $L_1$-embeddability of lamplighter graphs or groups, and more generally lamplighter metric spaces. Once this connection has been established, several new…

Fix $k \in \mathbb{N}$ and $0 < \delta < 1$. We study how large $N$ must be so that every $\delta$-dense subset $\mathcal{D} \subset \{0,1\}^N$ (meaning $|\mathcal{D}| \geq \delta 2^N$) contains the image of a metric embedding $f: \{0,1\}^k…

This article is a continuation of our article in [Canad. J. Math. Vol. 72 (3), (2020), pp. 774--804]. We construct orthogonal bases of the cycle and cut spaces of the Laakso graph $\mathcal{L}_n$. They are used to analyze projections from…

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…

It is shown that for every $\e\in (0,1)$, every compact metric space $(X,d)$ has a compact subset $S\subseteq X$ that embeds into an ultrametric space with distortion $O(1/\e)$, and $$\dim_H(S)\ge (1-\e)\dim_H(X),$$ where $\dim_H(\cdot)$…

We give a simple example of a countable metric space $M$ that does not embed bi-Lipschitz with distortion strictly less than 2 into any Asplund space. Actually, if $M$ embeds with distortion strictly less than 2 to a Banach space $X$, then…

Given a Banach space $X$ and a real number $\alpha\ge 1$, we write: (1) $D(X)\le\alpha$ if, for any locally finite metric space $A$, all finite subsets of which admit bilipschitz embeddings into $X$ with distortions $\le C$, the space $A$…

For every metric space $\mathcal X$ in which there exists a sequence of finite groups of bounded-size generating set that does not embed coarsely, and for every unbounded, increasing function $\rho$, we produce a group of subexponential…

Diversities are like metric spaces, except that every finite subset, instead of just every pair of points, is assigned a value. Just as there is a theory of minimal distortion embeddings of finite metric spaces into $L_1$, there is a…

By discretizing an argument of Kislyakov, Naor and Schechtman proved that the 1-Wasserstein metric over the planar grid $\{0,1,\dots n\}^2$ has $L_1$-distortion bounded below by a constant multiple of $\sqrt{\log n}$. We provide a new…

Metric embedding has become a common technique in the design of algorithms. Its applicability is often dependent on how high the embedding's distortion is. For example, embedding finite metric space into trees may require linear distortion…

In this paper we present new proofs of the non-embeddability of countably branching trees into Banach spaces satisfying property $(\beta_p)$ and of countably branching diamonds into Banach spaces which are $p$-AMUC for $p > 1$. These proofs…

In this article we start a systematic study of the bi-Lipschitz geometry of lamplighter graphs. We prove that lamplighter graphs over trees bi-Lipschitzly embed into Hamming cubes with distortion at most~$6$. It follows that lamplighter…