{
  "id": "2403.07900",
  "version": "v1",
  "published": "2024-02-25T19:44:14.000Z",
  "updated": "2024-02-25T19:44:14.000Z",
  "title": "Applications of equidistant supporting surfaces of a convex body in the hyperbolic space",
  "authors": [
    "Marek Lassak"
  ],
  "comment": "7 pages. arXiv admin note: text overlap with arXiv:2401.07831",
  "categories": [
    "math.MG"
  ],
  "abstract": "For a hyperplane $H$ supporting a convex body $C$ in the hyperbolic space $\\mathbb{H}^d$ we define the width of $C$ determined by $H$ as the distance between $H$ and a most distant ultraparallel hyperplane supporting $C$. The thickness (i.e., the minimum width) of $C$ is denoted by $\\Delta(C)$. A convex body $R \\subset \\mathbb{H}^d$ is called reduced if for every body $Z \\subsetneq R$ we have $\\Delta(Z) < \\Delta(R)$. We show that for any extreme point $e$ of a reduced body $R \\subset \\mathbb{H}^d$ there exists a supporting hyperplane $H$ of $R$ which passes through $e$ or its equidistant surface supporting $R$ passes through $e$. Bodies of constant width in $\\mathbb{H}^d$ are defined as bodies whose all widths are equal. We prove that every complete body in $\\mathbb{H}^d$ is a body of constant width.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-02-25T19:44:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52A55"
    ],
    "keywords": [
      "convex body",
      "equidistant supporting surfaces",
      "hyperbolic space",
      "constant width",
      "applications"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}