{
  "id": "1801.01161",
  "version": "v1",
  "published": "2018-01-03T20:44:24.000Z",
  "updated": "2018-01-03T20:44:24.000Z",
  "title": "Spherical bodies of constant width",
  "authors": [
    "Marek Lassak",
    "Michał Musielak"
  ],
  "categories": [
    "math.MG"
  ],
  "abstract": "The intersection $L$ of two different non-opposite hemispheres $G$ and $H$ of a $d$-dimensional sphere $S^d$ is called a lune. By the thickness of $L$ we mean the distance of the centers of the $(d-1)$-dimensional hemispheres bounding $L$. For a hemisphere $G$ supporting a %spherical convex body $C \\subset S^d$ we define ${\\rm width}_G(C)$ as the thickness of the narrowest lune or lunes of the form $G \\cap H$ containing $C$. If ${\\rm width}_G(C) =w$ for every hemisphere $G$ supporting $C$, we say that $C$ is a body of constant width $w$. We present properties of these bodies. In particular, we prove that the diameter of any spherical body $C$ of constant width $w$ on $S^d$ is $w$, and that if $w < \\frac{\\pi}{2}$, then $C$ is strictly convex. Moreover, we are checking when spherical bodies of constant width and constant diameter coincide.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-01-03T20:44:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52A55"
    ],
    "keywords": [
      "constant width",
      "spherical body",
      "constant diameter coincide",
      "dimensional sphere",
      "convex body"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}