Related papers: $(\beta)$-distortion of some infinite graphs

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…

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…

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…

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…

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 show that if the Szlenk index of a Banach space $X$ is larger than the first infinite ordinal $\omega$ or if the Szlenk index of its dual is larger than $\omega$, then the tree of all finite sequences of integers equipped with the…

We characterize the asymptotic behaviour of the compression associated to a uniform embedding into some Lp-space for a large class of groups including connected Lie groups with exponential growth and word-hyperbolic finitely generated…

The main result is a submetric characterization of the class of Banach spaces admitting an equivalent norm with Rolewicz's property ($\beta$). As applications we prove that up to renorming, property ($\beta$) is stable under coarse…

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 $A$…

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 note, we investigate the renorming theory of Banach spaces with property $(\beta)$ of Rolewicz. In particular, we give a "coordinate-free" proof of the fact that every Banach space with property $(\beta)$ admits an equivalent norm…

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…

Stochastic embeddings of finite metric spaces into graph-theoretic trees have proven to be a vital tool for constructing approximation algorithms in theoretical computer science. In the present work, we build out some of the basic theory of…

We prove an exponential separation in sample complexity between Euclidean and hyperbolic representations for learning on hierarchical data under standard Lipschitz regularization. For depth-$R$ hierarchies with branching factor $m$, we…

In this paper, we deal with a notion of Banach space-valued mappings defined on a set consisting of finite graphs with uniformly bounded vertex degree. These functions will be endowed with certain boundedness and additivity criteria. We…

In this note, we study some concentration properties for Lipschitz maps defined on Hamming graphs, as well as their stability under sums of Banach spaces. As an application, we extend a result of Causey on the coarse Lipschitz structure of…

This paper addresses the basic question of how well can a tree approximate distances of a metric space or a graph. Given a graph, the problem of constructing a spanning tree in a graph which strongly preserves distances in the graph is a…

The metric dimension reduction modulus $k^\alpha_n(\ell_\infty)$ is the smallest $k$ such that every $n$--point metric space can be embedded into some $k$-dimensional normed space, with bi--Lipschitz distortion at most $\alpha$. Determining…

We prove that a Banach space X is not super-reflexive if and only if the hyperbolic infinite tree embeds metrically into X. We improve one implication of J.Bourgain's result who gave a metrical characterization of super-reflexivity in…

In this paper we extend M. Lyubich's recent results on the global hyperbolicity of renormalization of quadratic-like germs to the space of C^r unimodal maps with quadratic critical point. We show that in this space the bounded-type limit…