Strongly non-embeddable metric spaces
type: Item
instance of: scholarly article
title: Strongly non-embeddable metric spaces
zbMATH Open document ID: 1242.46028
published in: Topology and its Applications
publication date: 2012-04-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/1109.5300
Mathematics Subject Classification ID: 46B85
Mathematics Subject Classification ID: 05C12
Mathematics Subject Classification ID: 46C05
Mathematics Subject Classification ID: 46T99
Mathematics Subject Classification ID: 54E35
zbMATH DE Number: 6023691
zbMATH Keywords: locally finite metric space
zbMATH Keywords: coarse embedding
zbMATH Keywords: uniform embedding
reviewed by: Mikhail Ostrovskii
MaRDI profile type: Publication
OpenAlex ID: W2070183896
arXiv ID: 1109.5300
cites work: Every separable metric space is Lipschitz equivalent to a subset of \(C_0\)
cites work: Uniform Embeddings into Hilbert Space and a Question of Gromov
cites work: On a problem of Smirnov
cites work: Q2737647
cites work: Generalized roundness and negative type
cites work: Coarse embeddings of metric spaces into Banach spaces
cites work: Q3415741
cites work: A theorem of Hrushovski-Solecki-Vershik applied to uniform and coarse embeddings of the Urysohn metric space
cites work: Q3568021
DOI: 10.1016/J.TOPOL.2011.11.041
Recommended article: Q3182189
Recommended article: Q5295717
Recommended article: Uniform Embeddings into Hilbert Space and a Question of Gromov
Recommended article: Q3062225
Recommended article: Embeddings of locally finite metric spaces into Banach spaces
Recommended article: COMPARING THE GENERALISED ROUNDNESS OF METRIC SPACES
Recommended article: TOPOLOGICAL STUDY OF GENERALIZED METRIC SPACES
Recommended article: Embeddings of proper metric spaces into Banach spaces
Recommended article: Coarse embeddings of metric spaces into Banach spaces
Recommended article: Q4518694
author: Casey Kelleher
author: Daniel Miller
author: Trenton Osborn
author: Anthony Weston
review text: Definition. Let \(p\geq 0\). A metric space \((X,d)\) is said to have \textit{generalized roundness} \(p\) if for all \(n\in\mathbb{N}\) and all \(a_1,\dots,a_n, b_1, \dots, b_n\in X\) the inequality NEWLINE\[NEWLINE\sum_{1\leq k<\ell\leq n}\left(d(a_k,a_\ell)^p+d(b_k,b_\ell)^p\right)\leq \sum_{1\leq j,i\leq n}d(a_j,b_i)^p NEWLINE\]NEWLINE holds. \textit{P. Enflo} [Ark. Mat. 8, 107--109 (1970; Zbl 0196.14003)] constructed a countable metric space that may not be uniformly embedded into any metric space of positive generalized roundness. \textit{A. N. Dranishnikov, G. Gong, V. Lafforgue} and \textit{G. Yu} [Can. Math. Bull. 45, No. 1, 60--70 (2002; Zbl 1044.46020)] modified Enflo's example to construct a locally finite metric space that may not be coarsely embedded into any Hilbert space. The authors of this paper combine these two examples and get a locally finite metric space which may not be embedded uniformly or coarsely into any metric space of non zero generalized roundness. It is important to mention that (1) by a uniform embedding the authors mean a uniformly continuous map \(f\) having uniformly continuous inverse \(f^{-1}\) defined on the image of \(f\); (2) in their definition of a locally finite metric space the authors do not require that \(\inf_{x\neq y}d(x,y)>0\).NEWLINENEWLINEThe authors make some related observations on Cayley graphs of infinite abelian groups and on Lipschitz injections.
description: scientific article; zbMATH DE number 6023691