# Necessary Condition for Rectifiability Involving Wasserstein Distance W2

@inproceedings{Dkabrowski2020NecessaryCF, title={Necessary Condition for Rectifiability Involving Wasserstein Distance W2}, author={Damian Dkabrowski}, year={2020} }

A Radon measure μ is n-rectifiable if μ ≪ H n and μ-almost all of supp μ can be covered by Lipschitz images of R. In this paper we give a necessary condition for rectifiability in terms of the so-called α2 numbers – coefficients quantifying flatness using Wasserstein distance W2. 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.

#### 3 Citations

Cones, rectifiability, and singular integral operators

- Mathematics
- Revista Matemática Iberoamericana
- 2021

Let μ be a Radon measure on R. We define and study conical energies Eμ,p(x, V, α), which quantify the portion of μ lying in the cone with vertex x ∈ R , direction V ∈ G(d, d−n), and aperture α ∈ (0,… Expand

Identifying 1-rectifiable measures in Carnot groups

- Mathematics
- 2021

We continue to develop a program in geometric measure theory that seeks to identify how measures in a space interact with canonical families of sets in the space. In particular, extending a theorem… Expand

An $\alpha$-number characterization of $L^{p}$ spaces on uniformly rectifiable sets

- Mathematics
- 2020

We give a characterization of $L^{p}(\sigma)$ for uniformly rectifiable measures $\sigma$ using Tolsa's $\alpha$-numbers, by showing, for $1<p<\infty$ and $f\in L^{p}(\sigma)$, that
\[
\lVert… Expand

#### References

SHOWING 1-10 OF 49 REFERENCES

Multiscale Analysis of 1-rectifiable Measures II: Characterizations

- Mathematics
- 2016

Abstract A measure is 1-rectifiable if there is a countable union of finite length curves whose complement has zero measure. We characterize 1-rectifiable Radon measures μ in n-dimensional Euclidean… Expand

Characterization of rectifiable measures in terms of 𝛼-numbers

- Mathematics
- 2018

We characterize Radon measures $\mu$ in $\mathbb{R}^{n}$ that are $d$-rectifiable in the sense that their supports are covered up to $\mu$-measure zero by countably many $d$-dimensional Lipschitz… Expand

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

- Mathematics
- 2019

A Radon measure $\mu$ is $n$-rectifiable if it is absolutely continuous with respect to $\mathcal{H}^n$ and $\mu$-almost all of $\text{supp}\,\mu$ can be covered by Lipschitz images of… Expand

Generalized rectifiability of measures and the identification problem

- Mathematics
- Complex Analysis and its Synergies
- 2019

One goal of geometric measure theory is to understand how measures in the plane or a higher dimensional Euclidean space interact with families of lower dimensional sets. An important dichotomy arises… Expand

Analysis of and on uniformly rectifiable sets

- Mathematics
- 1993

The notion of uniform rectifiability of sets (in a Euclidean space), which emerged only recently, can be viewed in several different ways. It can be viewed as a quantitative and scale-invariant… Expand

Uniform rectifiability, Calderón–Zygmund operators with odd kernel, and quasiorthogonality

- Mathematics
- 2009

In this paper we study some questions in connection with uniform rectifiability and the $L^2$ boundedness of Calderon-Zygmund operators. We show that uniform rectifiability can be characterized in… Expand

Characterization of n-rectifiability in terms of Jones’ square function: Part II

- Mathematics
- 2015

We show that a Radon measure $${\mu}$$μ in $${\mathbb{R}^d}$$Rd which is absolutely continuous with respect to the n-dimensional Hausdorff measure $${\mathcal{H}^n}$$Hn is n-rectifiable if the so… Expand

Rectifiability of measures and the $\beta_p$ coefficients

- Mathematics
- Publicacions Matemàtiques
- 2019

In some former works of Azzam and Tolsa it was shown that $n$-rectifiability can be characterized in terms of a square function involving the David-Semmes $\beta_2$ coefficients. In the present paper… Expand

Rectifiable sets and the Traveling Salesman Problem

- Mathematics
- 1990

Let K c C be a bounded set. In this paper we shall give a simple necessary and sufficient condit ion for K to lie in a rectifiable curve. We say that a set is a rectifiable curve if it is the image… Expand

A doubling measure on R^d can charge a rectifiable curve

- Mathematics
- 2009

For d > 2, we construct a doubling measure v on ℝ d and a rectifiable curve Γ such that ν(Γ) > 0.