{
  "id": "1408.6645",
  "version": "v1",
  "published": "2014-08-28T08:18:54.000Z",
  "updated": "2014-08-28T08:18:54.000Z",
  "title": "Wasserstein Distance and the Rectifiability of Doubling Measures: Part I",
  "authors": [
    "Jonas Azzam",
    "Guy David",
    "Tatiana Toro"
  ],
  "comment": "85 pages, 2 figures",
  "categories": [
    "math.CA"
  ],
  "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 $\\int_0^1 \\alpha(x,r) \\frac{dr}{r} < \\infty$ 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. Soit $\\mu$ une mesure doublante dans $\\mathbb{R}^n$. On \\'etudie des relations quantifi\\'ees entre la rectifiabilit\\'e de $\\mu$ et la distance entre $\\mu$ et les mesures plates. Plus pr\\'ecis\\'ement, on utilise une variante de la $L^1$-distance de Wasserstein pour d\\'efinir, pour $x$ dans le support $\\Sigma$ de $\\mu$ et $r>0$, un nombre $\\alpha(x,r)$ qui mesure la distance minimale entre la restriction de $\\mu$ \\`a $B(x,r)$ et une mesure de Lebesgue sur un sous-espace affine passant par $B(x,r/2)$. On d\\'ecompose l'ensemble des points $x\\in \\Sigma$ tels que $\\int_0^1 \\alpha(x,r) \\frac{dr}{r} < \\infty$ en parties rectifiables de dimensions diverses, et on obtient un meilleur contr\\^ole de ces parties et de la taille de $\\mu$ quand les $\\alpha(x,r)$ v\\'erifient certaines conditions de Carleson.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-08-28T08:18:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A75",
      "28A78"
    ],
    "keywords": [
      "wasserstein distance",
      "doubling measure",
      "rectifiability",
      "sous-espace affine passant par",
      "carleson measure estimates hold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 85,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1408.6645A"
    }
  }
}