Necessary condition for rectifiability involving Wasserstein distance $W_2$

A Radon measure $\mu$ is $n$-rectifiable if $\mu \ll {\mathcal{H}}^n$ and $\mu$-almost all of $\text{supp}\mu$ 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.