{
  "id": "2401.16207",
  "version": "v1",
  "published": "2024-01-29T15:01:49.000Z",
  "updated": "2024-01-29T15:01:49.000Z",
  "title": "Probability that $n$ points are in convex position in a regular $κ$-gon : Asymptotic results",
  "authors": [
    "Ludovic Morin"
  ],
  "comment": "51 pages, 18 figures, 4 pictures",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "Let $\\mathbb{P}_{\\kappa}(n)$ be the probability that $n$ points $z_1,\\ldots,z_n$ picked uniformly and independently in $\\mathfrak{C}_\\kappa$, a regular $\\kappa$-gon with area $1$, are in convex position, that is, form the vertex set of a convex polygon. In this paper, we give an equivalent of $\\mathbb{P}_{\\kappa}(n)$ for all $\\kappa\\geq 3$, which improves on a famous result of B\\'ar\\'any. A second aim of the paper is to establish a limit theorem which describes the fluctuations around the limit shape of a $n$-tuple of points in convex position when $n\\to+\\infty$. Finally, we give an algorithm asymptotically exact for the random generation of $z_1,\\ldots,z_n$, conditioned to be in convex position in $\\mathfrak{C}_\\kappa$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-29T15:01:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52A22",
      "60D05"
    ],
    "keywords": [
      "convex position",
      "asymptotic results",
      "probability",
      "second aim",
      "convex polygon"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 51,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}