Necessary condition for rectifiability involving Wasserstein distance $W_2$

Abstract

A Radon measure $\mu$ is $n$-rectifiable if it is absolutely continuous with respect to $n$-dimensional Hausdorff measure and $\mu$-almost all of supp$\mu$ can be covered by Lipschitz images of ${\mathbb{R}}^n$⁠. In this paper, we give a necessary condition for rectifiability in terms of the so-called ${\alpha }_2$ numbers — coefficients quantifying flatness using Wasserstein distance $W_2$⁠. In a recent article, we showed that the same condition is also sufficient for rectifiability, and so we get a new characterization of rectifiable measures.