{
  "id": "2109.06545",
  "version": "v1",
  "published": "2021-09-14T09:31:28.000Z",
  "updated": "2021-09-14T09:31:28.000Z",
  "title": "A non-existence result for the $L_p$-Minkowski problem",
  "authors": [
    "Christos Saroglou"
  ],
  "comment": "10 pages",
  "categories": [
    "math.MG"
  ],
  "abstract": "We show that given a real number $p<1$, a positive integer $n$ and a proper subspace $H$ of $\\mathbb{R}^n$, the measure on the Euclidean sphere $\\mathbb{S}^{n-1}$, which is concentrated in $H$ and whose restriction to the class of Borel subsets of $\\mathbb{S}^{n-1}\\cap H$ equals the spherical Lebesgue measure on $\\mathbb{S}^{n-1}\\cap H$, is not the $L_p$-surface area measure of any convex body. This, in particular, disproves a conjecture from [Bianchi, B\\\"or\\\"oczky, Colesanti, Yang, The $L_p$-Minkowski problem for $-n<p<1$, Adv. Math. (2019)].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-14T09:31:28.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "minkowski problem",
      "non-existence result",
      "surface area measure",
      "borel subsets",
      "proper subspace"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}