Characterizing rectifiability via biLipschitz pieces of Lipschitz mappings on the space

2025/10/15 by Sean Li, Li, Sean, Raanan Schul +1 · 1 citation
Mathematics · Computer Science · #Advanced Topology and Set Theory #Optimization and Variational Analysis #Advanced Banach Space Theory

paper · pdf · doi:10.48550/arxiv.2510.13525

Abstract

We give the following characterization of rectifiable metric spaces. A metric space with positive lower Hausdorff density is rectifiable if and only if, for any subset F and f:F→ Y, a Lipschitz map into a metric space with positive measure image (of the same dimension), there exists a positive measure subset A⊂ F so that f is biLipschitz on A. We also give a characterization in terms of a full biLipschitz decomposition. These characterizations are new even for subsets of Euclidean space. One of our tools is Alberti representations. On the way we give a method for constructing independent Alberti representations, which may be of independent interest. We use this to characterize unrectifiable metric spaces as those spaces for which there exist a positive measure subset S and a Lipschitz map ϕ into a lower dimensional Euclidean space so that S is \cH¹-null with respect to all curve fragments that are quantitatively transversal to ϕ.

Citations

• Alberti representations, rectifiability of metric spaces and higher integrability of measures satisfying a PDE
• Ahlfors-regular distances on the Heisenberg group without biLipschitz pieces

Cited by

• The Heinonen-Semmes problems after thirty years

Related

• Uniformly rectifiable metric spaces: Lipschitz images, Bi-Lateral Weak Geometric Lemma and Corona Decompositions
• Connections between metric differentiability and rectifiability
• Carnot rectifiability and Alberti representations
• Bi-Lipschitz Decomposition of Lipschitz functions into a Metric space
• Alberti representations, rectifiability of metric spaces and higher integrability of measures satisfying a PDE