Lipschitz mappings, metric differentiability, and factorization through metric trees

TL;DR

This paper characterizes when Lipschitz maps into metric spaces factor through metric trees, proves a conjecture relating topological dimension and Hausdorff measure, and introduces an area formula for length-preserving maps in metric spaces.

Contribution

It provides new equivalent conditions for Lipschitz factorization through metric trees and proves a conjecture linking topological dimension with Hausdorff measure.

Findings

Lipschitz maps from cubes factor through metric trees under certain conditions

Topological dimension equals Hausdorff measure for Lipschitz images of open sets in Euclidean space

New area formula for length-preserving maps in metric spaces

Abstract

Given a Lipschitz map $f$ from a cube into a metric space, we find several equivalent conditions for $f$ to have a Lipschitz factorization through a metric tree. As an application we prove a recent conjecture of David and Schul. The techniques developed for the proof of the factorization result yield several other new and seemingly unrelated results. We prove that if $f$ is a Lipschitz mapping from an open set in $\mathbb{R}^n$ onto a metric space $X$, then the topological dimension of $X$ equals $n$ if and only if $X$ has positive $n$-dimensional Hausdorff measure. We also prove an area formula for length-preserving maps between metric spaces, which gives, in particular, a new formula for integration on countably rectifiable sets in the Heisenberg group.