On the nonexistence of bilipschitz parameterizations and geometric problems about \(A_ \infty\)-weights
type: Item
instance of: scholarly article
title: On the nonexistence of bilipschitz parameterizations and geometric problems about \(A_ \infty\)-weights
zbMATH Open document ID: 0858.46017
DOI: 10.4171/RMI/201
author: Stephen Semmes
published in: Revista Matemática Iberoamericana
publication date: 1996-10-10T00:00:00Z
full work available at URL: https://eudml.org/doc/39505
review text: Summary: How can one recognize when a metric space is bi-Lipschitz equivalent to a Euclidean space? One should not take that abstraction of metric spaces too seriously here; subsets of \(\mathbb{R}^n\) are already quite interesting. It is easy to generate geometric conditions which are necessary for bi-Lipschitz equivalence, but it is not clear that such conditions should ever be sufficient. The main point of this paper is that the optimistic conjectures about the existence of bi-Lipschitz parameterizations are wrong. In other words, there are spaces whose geometry is very similar to but still distinct from Euclidean geometry. Related questions of bi-Lipschitz equivalence and embeddings are addressed for metric spaces obtained by deforming the Euclidean metric on \(\mathbb{R}^n\) using an \(A_\infty\) weight.
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 46B07
zbMATH DE Number: 933288
zbMATH Keywords: bi-Lipschitz equivalent to a Euclidean space
zbMATH Keywords: bi-Lipschitz parameterizations
zbMATH Keywords: deforming the Euclidean metric
zbMATH Keywords: \(A_ \infty\) weight
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 933288