{
  "id": "1202.0045",
  "version": "v4",
  "published": "2012-01-31T22:43:00.000Z",
  "updated": "2015-03-10T09:03:46.000Z",
  "title": "Shortest Path through Random Points",
  "authors": [
    "Sung Jin Hwang",
    "Steven B. Damelin",
    "Alfred O. Hero III"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $(M,g_1)$ be a complete $d$-dimensional Riemannian manifold for $d > 1$. Let $\\mathcal X_n$ be a set of $n$ sample points in $M$ drawn randomly from a smooth Lebesgue density $f$ supported in $M$. Let $x,y$ be two points in $M$. We prove that the normalized length of the power-weighted shortest path between $x, y$ through $\\mathcal X_n$ converges almost surely to a constant multiple of the Riemannian distance between $x,y$ under the metric tensor $g_p = f^{2(1-p)/d} g_1$, where $p > 1$ is the power parameter.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-03-23T17:06:14.000Z",
      "abstract": "Let $(M,g_1)$ be a complete $d$-dimensional Riemannian manifold for $d > 1$. Let $\\mathcal X_n$ be a set of $n$ sample points in $M$ drawn randomly from a smooth Lebesgue density $f$ supported in $M$. Let $x,y$ be two points in $M$. We prove that the normalized length of the power-weighted shortest path between $x, y$ through $\\mathcal X_n$ converges almost surely to a constant multiple of the Riemannian distance between $x,y$ under the metric $g_p = f^{2(1-p)/d} g_1$, where $p > 1$ is the power parameter.",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v4",
      "updated": "2015-03-10T09:03:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F15",
      "60C05",
      "53B21"
    ],
    "keywords": [
      "random points",
      "dimensional riemannian manifold",
      "smooth lebesgue density",
      "constant multiple",
      "power-weighted shortest path"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1202.0045H"
    }
  }
}