Wasserstein distance and the rectifiability of doubling measures: part I

Abstract

Let μ be a doubling measure in R n . We investigate quantitative relations between the rectifiability of μ and its distance to flat measures. More precisely, for x in the support of μ and r > 0, we introduce a number α(x, r ) ∈ (0, 1] that measures, in terms of a variant of the L 1 -Wasserstein distance, the minimal distance between the restriction of μ 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 wheré 1 0 α(x, r ) dr r < ∞ can be decomposed into rectifiable pieces of various dimensions. We obtain additional control on the pieces and the size of μ when we assume that some Carleson measure estimates hold. Résumé en FrançaisSoit μ une mesure doublante dans R n . On étudie des relations quantifiées entre la rectifiabilité de μ et la distance entre μ et les mesures plates. Plus précisément, on utilise une variante de la L 1 -distance de Wasserstein pour définir, pour x dans le support de μ et r > 0, un nombre α(x, r ) qui mesure la distance minimale entre la restriction de μ à B(x, r ) et une mesure de Lebesgue sur un sousespace affine passant par B(x, r/2). On décompose l'ensemble des points x ∈ tels que´1 0 α(x, r ) dr r < ∞ en parties rectifiables de dimensions diverses, et on obtient un meilleur contrôle de ces parties et de la taille de μ quand les α(x, r ) vérifient certaines conditions de Carleson.

Multiscale Analysis of 1-rectifiable Measures II: Characterizations

“…see [13] and [14]). Much work has been done in connecting beta numbers and recti ability (highlights include [50], [32], [33], [24], [34], [15], [5], [2], [6], [46]), but we single out two recent papers, [58] and [4], and state one theorem, which is a combination of their main results. …”

Characterization of rectifiable measures in terms of 𝛼-numbers

“…x ∈ R n , then µ is d-rectifiable. In [ADT16] it was also conjectured that the same result should be true if α d µ (x, r) were replaced with α d µ (x, r) 2 . In [Orp17], Orponen showed the conjecture is true for n = d = 1.…”

“…It is not hard to see using the definition of Wasserstein distance that α d µ (x, r) ≤ α d µ (x, r), and so Theorem I implies the conjecture from [ADT16] for measures satisfying (1.9).…”

“…In [ADT16], the first author, David and the third author considered some variant of the α coefficients. Denote T x,r (y) = (y − x)/r and let W 1 be the 1-Wasserstein distance between probability measures and the infimum is taken over all d-planes.…”

“…Then one sets where the infimum is taken over all d-planes. In [ADT16] it was shown that if µ is doubling and 1 0 α d µ (x, r) dr r < ∞ for µ-a.e. x ∈ R n , then µ is d-rectifiable.…”