{
  "id": "2306.04412",
  "version": "v1",
  "published": "2023-06-07T13:13:34.000Z",
  "updated": "2023-06-07T13:13:34.000Z",
  "title": "Width of convex bodies in hyperbolic space",
  "authors": [
    "Marek Lassak"
  ],
  "comment": "13 pages, 6 figures",
  "categories": [
    "math.MG"
  ],
  "abstract": "For every 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 $C$. We define bodies of constant width in $\\mathbb{H}^d$ in the standard way as bodies whose all widths are equal. We show that every body of constant width is strictly convex. The minimum width of $C$ over all supporting $H$ is called the thickness $\\Delta (C)$ of $C$. A convex body $R \\subset \\mathbb{H}^d$ is said to be reduced if $\\Delta (Z) < \\Delta (R)$ for every convex body $Z$ properly contained in $R$. We show that regular tetrahedra in $\\mathbb{H}^3$ are not reduced. Similarly as in $\\mathbb{E}^d$, we introduce complete bodies and bodies of constant diameter. They appear to coincide with bodies of constant width of the same diameter.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-06-07T13:13:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52A55",
      "A.0"
    ],
    "keywords": [
      "convex body",
      "hyperbolic space",
      "constant width",
      "distant ultraparallel hyperplane supporting",
      "define bodies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}