On the Lipschitz dimension of Cheeger–Kleiner
Tom 253 / 2021
Fundamenta Mathematicae 253 (2021), 317-358
MSC: Primary 30L99; Secondary 30L05, 54F45, 53C23, 51F30.
DOI: 10.4064/fm776-8-2020
Opublikowany online: 4 November 2020
Streszczenie
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.
