Characterizing rectifiability via biLipschitz pieces of Lipschitz mappings on the space

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 $f:F\to Y$, a Lipschitz map into a metric space with positive measure image (of the same dimension), there exists a positive measure subset $A\subset F$ 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 $\phi$ into a lower dimensional Euclidean space so that $S$ is $\cH^1$-null with respect to all curve fragments that are quantitatively transversal to $\phi$.