# Necessary condition for rectifiability involving Wasserstein distance $W_2$

## TL;DR

This paper establishes a necessary condition for the rectifiability of Radon measures using Wasserstein distance-based flatness coefficients, complementing previous sufficiency results to characterize rectifiable measures.

## Contribution

It introduces a new necessary condition for rectifiability involving Wasserstein distance, completing a characterization when combined with prior sufficiency results.

## Key findings

- Necessary condition for rectifiability via $eta_2$ coefficients
- Characterization of rectifiable measures using Wasserstein distance
- Bridging flatness coefficients with geometric measure theory

## Abstract

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.