# On the Lipschitz dimension of Cheeger-Kleiner

## TL;DR

This paper explores the properties of Lipschitz dimension in metric spaces, examining its behavior under convergence, mappings, and its relation to other dimensions, with computations for various classical spaces.

## Contribution

It provides a comprehensive analysis of Lipschitz dimension, including its invariance properties, relationships to other dimensions, and explicit calculations for key examples.

## Key findings

- Lipschitz dimension is not invariant under all mappings.
- Computed Lipschitz dimension for Carnot groups, snowflakes, trees, and Sierpinski carpets.
- Derived non-embedding results and conditions for Lipschitz maps.

## 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 Sierpinski 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.