Sufficient Condition for Rectifiability Involving Wasserstein Distance $$W_2$$

A Radon measure \(\mu \) is n-rectifiable if it absolutely continuous with respect to \({\mathcal {H}}^n\) and \)-almost all of \({{\,\mathrm{supp}\,}}\mu can be covered by Lipschitz images \({\mathbb {R}}^n\). In this paper we give two sufficient conditions for rectifiability, both in terms square functions flatness-quantifying coefficients. The first condition involves the so-called \(\alpha \(\beta _2\) numbers. second one numbers—coefficients quantifying flatness via Wasserstein distance \(W_2\). Both are necessary too—the was shown Tolsa, while necessity established our recent paper. Thus, get new characterizations rectifiability.