{
  "id": "2406.06771",
  "version": "v1",
  "published": "2024-06-10T20:14:40.000Z",
  "updated": "2024-06-10T20:14:40.000Z",
  "title": "On Extremal Problems Associated with Random Chords on a Circle",
  "authors": [
    "Cynthia Bortolotto",
    "João P. G. Ramos"
  ],
  "comment": "23 pages, 3 figures",
  "categories": [
    "math.MG",
    "math.PR"
  ],
  "abstract": "Inspired by the work of Karamata, we consider an extremization problem associated with the probability of intersecting two random chords inside a circle of radius $r, \\, r \\in (0,1]$, where the endpoints of the chords are drawn according to a given probability distribution on $\\mathbb{S}^1$. We show that, for $r=1,$ the problem is degenerated in the sense that any continuous measure is an extremiser, and that, for $r$ sufficiently close to $1,$ the desired maximal value is strictly below the one for $r=1$ by a polynomial factor in $1-r.$ Finally, we prove, by considering the auxiliary problem of drawing a single random chord, that the desired maximum is $1/4$ for $r \\in (0,1/2).$ Connections with other variational problems and energy minimization problems are also presented.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-10T20:14:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "extremal problems",
      "energy minimization problems",
      "random chords inside",
      "single random chord",
      "variational problems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}