2021/06/30 by Behnam Esmayli, Piotr Hajłasz, Esmayli, Behnam +1
Computer Science · Mathematics · #30L99 #51F30 #53C17 #53C23 #54F45 #54F50 #Advanced Topology and Set Theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Dynamics and Fractals #Metric Geometry (math.MG) #Primary: 28A75 #Secondary: 28A78 #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2106.15763
openalex publication_date 2021/06/30 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
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 ℝn 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.