{
  "id": "1801.08008",
  "version": "v1",
  "published": "2018-01-24T14:53:29.000Z",
  "updated": "2018-01-24T14:53:29.000Z",
  "title": "Cones generated by random points on half-spheres and convex hulls of Poisson point processes",
  "authors": [
    "Zakhar Kabluchko",
    "Alexander Marynych",
    "Daniel Temesvari",
    "Christoph Thaele"
  ],
  "categories": [
    "math.PR",
    "math.MG"
  ],
  "abstract": "Let $U_1,U_2,\\ldots$ be random points sampled uniformly and independently from the $d$-dimensional upper half-sphere. We show that, as $n\\to\\infty$, the $f$-vector of the $(d+1)$-dimensional convex cone $C_n$ generated by $U_1,\\ldots,U_n$ weakly converges to a certain limiting random vector, without any normalization. We also show convergence of all moments of the $f$-vector of $C_n$ and identify the limiting constants for the expectations. We prove that the expected Grassmann angles of $C_n$ can be expressed through the expected $f$-vector. This yields convergence of expected Grassmann angles and conic intrinsic volumes and answers thereby a question of B\\'ar\\'any, Hug, Reitzner and Schneider [Random points in halfspheres, Rand. Struct. Alg., 2017]. Our approach is based on the observation that the random cone $C_n$ weakly converges, after a suitable rescaling, to a random cone whose intersection with the tangent hyperplane of the half-sphere at its north pole is the convex hull of the Poisson point process with power-law intensity function proportional to $\\|x\\|^{-(d+\\gamma)}$, where $\\gamma=1$. We compute the expected number of facets, the expected intrinsic volumes and the expected $T$-functional of this random convex hull for arbitrary $\\gamma>0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-01-24T14:53:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52A22",
      "60D05",
      "52A55",
      "52B11",
      "60F05"
    ],
    "keywords": [
      "poisson point process",
      "random points",
      "expected grassmann angles",
      "random cone",
      "power-law intensity function proportional"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}