Necessary condition for rectifiability involving Wasserstein distance $W_2$

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.

27 pages; added Remarks 1.4, 1.5, 1.7; added Subsection 2.4 about parameters appearing in the proof; moved the "second approximation argument" from Section 4 to the end of Section 3; other small fixes and improvements; results unchanged

References in corpus (4)

• Necessary condition for rectifiability involving Wasserstein distance $W_2$
• Quantitative Reifenberg theorem for measures
• Sufficient condition for rectifiability involving Wasserstein distance $W_2$
• Absolute continuity and $\alpha$-numbers on the real line

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