{
  "id": "2107.05402",
  "version": "v1",
  "published": "2021-07-12T13:06:35.000Z",
  "updated": "2021-07-12T13:06:35.000Z",
  "title": "The Duality of the Volumes and the Numbers of Vertices of Random Polytopes",
  "authors": [
    "Christian Buchta"
  ],
  "comment": "8 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "An identity due to Efron dating from 1965 relates the expected volume of the convex hull of $n$ random points to the expected number of vertices of the convex hull of $n+1$ random points. Forty years later this identity was extended from expected values to higher moments. The generalized identity has attracted considerable interest. Whereas the left-hand side of the generalized identity -- concerning the volume -- has an immediate geometric interpretation, this is not the case for the right-hand side -- concerning the number of vertices. A transformation of the right-hand side applying an identity for elementary symmetric polynomials overcomes the blemish. The arising formula reveals a duality between the volumes and the numbers of vertices of random polytopes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-12T13:06:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60D05",
      "52A22"
    ],
    "keywords": [
      "random polytopes",
      "convex hull",
      "random points",
      "right-hand side",
      "elementary symmetric polynomials overcomes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}