Lipschitz mappings, metric differentiability, and factorization through metric trees

Abstract

Given a Lipschitz map f $f$ from a cube into a metric space, we find several equivalent conditions for f $f$ 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 $f$ is a Lipschitz mapping from an open set in R n $\mathbb {R}^n$ onto a metric space X $X$ , then the topological dimension of X $X$ equals n $n$ if and only if X $X$ has positive n $n$ -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.