Metric differentiation, monotonicity and maps to \(L^{1}\)
type: Item
instance of: scholarly article
title: Metric differentiation, monotonicity and maps to \(L^{1}\)
zbMATH Open document ID: 1214.46013
author: Jeff Cheeger
author: Bruce Kleiner
published in: Inventiones Mathematicae
publication date: 2010-11-23T00:00:00Z
full work available at URL: https://arxiv.org/abs/0907.3295
review text: The authors study Lipschitz embeddings in the category of topological spaces. As is well known, Lipschitz continuity is stronger than continuity, since, roughly speaking, it requires bounded variations. More precisely, if \(f:X\to Y\) is a mapping between metric spaces \((X,d_X)\) and \((Y,d_Y)\), then \(f\) is Lipschitz continuous iff there exists a constant \(K\geq 0\) such that for all \(x_1\not= x_2\), \(d_Y(f(x_1),f(x_2))/d_X(x_1,x_2)\leq K\). Furthermore, \(f\) is called bi-Lipschitz if there exists a \(K\geq 1\) such that \({{1}\over{K}}d_X(x_1,x_2)\leq d_Y(f(x_1),f(x_2))\leq K d_X(x_1,x_2)\) for all \(x_1\not= x_2\). A surjective bi-Lipschitz mapping is an isomorphism in the Lipschitz category of metric spaces (where morphisms are Lipschitz maps). For a bijective, bi-Lipschitz mapping \(f\), the inverse function is also bi-Lipschitz with the same constant \(K\) of \(f\). It is interesting to emphasize that a differentiable function \(f:{\mathbb R}\to{\mathbb R}\) is not necessarily Lipschitz. Conversely, a Lipschitz function \(g:{\mathbb R}\to{\mathbb R}\) is absolutely continuous and therefore is differentiable almost everywhere. Furthermore \(\|Df(x)\|\leq K\), where \(K\) is the Lipschitz constant of \(f\) (Rademacher's theorem). Rademacher's theorem extends to Lipschitz functions \(f:X\to{\mathbb R}\), where \((X,\mu)\) is a doubling measure metric space satisfying a Poincaré inequality (Cheeger's theorem). A metric space is doubling if there is a constant \(N\) such that any ball can be covered by at most \(N\) balls of half the radius. If \((X,d)\) is a doubling metric space and \(0<\alpha<1\), then there is a bi-Lipschitz embedding \(f:(X,d^\alpha)\to{\mathbb R}^n\) (Assouad's theorem). By the concept of bi-Lipschitz mappings, it is possible to recognize in the category of topological manifolds a little subcategory of Lipschitz manifolds, where the Lipschitz structure is defined by means of a pseudogroup of bi-Lipschitz homeomorphisms. A PL structure …
reviewed by: Agostino Prástaro
Mathematics Subject Classification ID: 46B85
Mathematics Subject Classification ID: 46B80
Mathematics Subject Classification ID: 57N20
Mathematics Subject Classification ID: 22F30
Mathematics Subject Classification ID: 22E05
Mathematics Subject Classification ID: 30L05
zbMATH DE Number: 5818341
zbMATH Keywords: global analysis in infinite-dimensional manifolds
zbMATH Keywords: bi-Lipschitz embeddings
zbMATH Keywords: Heisenberg group
zbMATH Keywords: Carathéodory metric
zbMATH Keywords: cut metrics
MaRDI profile type: Publication
OpenAlex ID: W3104961280
arXiv ID: 0907.3295
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DOI: 10.1007/S00222-010-0264-9
description: scientific article