# Sufficient condition for rectifiability involving Wasserstein distance $W_2$

## TL;DR

This paper provides two new sufficient conditions for the rectifiability of Radon measures, using square functions of flatness coefficients involving Wasserstein distance $W_2$, and establishes their necessity, offering new characterizations.

## Contribution

The paper introduces two novel sufficient conditions for rectifiability based on flatness coefficients, one involving Wasserstein distance $W_2$, and proves their necessity, enriching the understanding of measure rectifiability.

## Key findings

- Both conditions are necessary and sufficient for rectifiability.
- The $eta_2$ and $ ext{α}$ coefficients characterize rectifiability.
- The $ ext{α}_2$ coefficients involving Wasserstein distance $W_2$ provide a new characterization.

## Abstract

A Radon measure $\mu$ is $n$-rectifiable if it is absolutely continuous with respect to $\mathcal{H}^n$ and $\mu$-almost all of $\text{supp}\,\mu$ can be covered by Lipschitz images of $\mathbb{R}^n$. In this paper we give two sufficient conditions for rectifiability, both in terms of square functions of flatness-quantifying coefficients. The first condition involves the so-called $\alpha$ and $\beta_2$ numbers. The second one involves $\alpha_2$ numbers -- coefficients quantifying flatness via Wasserstein distance $W_2$. Both conditions are necessary for rectifiability, too -- the first one was shown to be necessary by Tolsa, while the necessity of the $\alpha_2$ condition is established in our recent paper. Thus, we get two new characterizations of rectifiability.