Sufficient Condition for Rectifiability Involving Wasserstein Distance W2

A Radon measure mu is n-rectifiable if it is absolutely continuous with respect to H-n and mu-almost all of supp mu can be covered by Lipschitz images of 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 viaWasserstein 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 a2 condition is established in our recent paper. Thus, we get two newcharacterizations of rectifiability.