Wasserstein distance and the rectifiability of doubling measures: part I

Abstract

Let $$\mu $$ be a doubling measure in $${\mathbb {R}}^n$$ . We investigate quantitative relations between the rectifiability of $$\mu $$ and its distance to flat measures. More precisely, for $$x$$ in the support $$\Sigma $$ of $$\mu $$ and $$r > 0$$ , we introduce a number $$\alpha (x,r)\in (0,1]$$ that measures, in terms of a variant of the $$L^1$$ -Wasserstein distance, the minimal distance between the restriction of $$\mu $$ to $$B(x,r)$$ and a multiple of the Lebesgue measure on an affine subspace that meets $$B(x,r/2)$$ . We show that the set of points of $$\Sigma $$ where $$\int _0^1 \alpha (x,r) {dr \over r} < \infty $$ can be decomposed into rectifiable pieces of various dimensions. We obtain additional control on the pieces and the size of $$\mu $$ when we assume that some Carleson measure estimates hold.