{
  "id": "1904.11004",
  "version": "v1",
  "published": "2019-04-24T18:37:01.000Z",
  "updated": "2019-04-24T18:37:01.000Z",
  "title": "Sufficient condition for rectifiability involving Wasserstein distance $W_2$",
  "authors": [
    "Damian Dąbrowski"
  ],
  "comment": "53 pages",
  "categories": [
    "math.CA",
    "math.AP"
  ],
  "abstract": "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 $\\mathbb{R}^n$. 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 via Wasserstein 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 $\\alpha_2$ condition is established in our recent paper. Thus, we get two new characterizations of rectifiability.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-24T18:37:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A75",
      "28A78"
    ],
    "keywords": [
      "wasserstein distance",
      "sufficient condition",
      "rectifiability",
      "lipschitz images",
      "radon measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 53,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}