{
  "id": "2410.11706",
  "version": "v1",
  "published": "2024-10-15T15:44:32.000Z",
  "updated": "2024-10-15T15:44:32.000Z",
  "title": "Probability that $n$ points are in convex position in a general convex polygon: Asymptotic results",
  "authors": [
    "Ludovic Morin"
  ],
  "comment": "21 pages, 19 figures",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "Let $\\mathbb{P}_K(n)$ be the probability that $n$ points $z_1,\\ldots,z_n$ picked uniformly and independently in $K$, a non-flat compact convex polygon in $\\mathbb{R}^2$, are in convex position, that is, form the vertex set of a convex polygon. In this paper, we give an equivalent of $\\mathbb{P}_K(n)$ when $n\\to\\infty$. This improves on a famous result of B\\'ar\\'any (yet valid for a general convex domain $K$) and a result we initiated in the case where $K$ is a regular convex polygon.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-15T15:44:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52A22",
      "60D05"
    ],
    "keywords": [
      "general convex polygon",
      "convex position",
      "asymptotic results",
      "probability",
      "non-flat compact convex polygon"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}