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On the Lipschitz dimension of Cheeger–Kleiner

Volume 253 / 2021

Guy C. David Fundamenta Mathematicae 253 (2021), 317-358 MSC: Primary 30L99; Secondary 30L05, 54F45, 53C23, 51F30. DOI: 10.4064/fm776-8-2020 Published online: 4 November 2020

Abstract

In a 2013 paper, Cheeger and Kleiner introduced a new type of dimension for metric spaces, the “Lipschitz dimension”. We study the dimension-theoretic properties of Lipschitz dimension, including its behavior under Gromov–Hausdorff convergence, its (non-)invariance under various classes of mappings, and its relationship to the Nagata dimension and Cheeger’s “analytic dimension”. We compute the Lipschitz dimension of various natural spaces, including Carnot groups, snowflakes of Euclidean spaces, metric trees, and Sierpiński carpets. As corollaries, we obtain a short proof of a quasi-isometric non-embedding result for Carnot groups and a necessary condition for the existence of non-degenerate Lipschitz maps between certain spaces.

Authors

  • Guy C. DavidDepartment of Mathematical Sciences
    Ball State University
    Muncie, IN 47306, U.S.A.
    e-mail

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