{
  "id": "1404.7019",
  "version": "v1",
  "published": "2014-04-28T15:24:02.000Z",
  "updated": "2014-04-28T15:24:02.000Z",
  "title": "Typical curvature behaviour of bodies of constant width",
  "authors": [
    "Imre Barany",
    "rolf Schneider"
  ],
  "categories": [
    "math.MG"
  ],
  "abstract": "It is known that an $n$-dimensional convex body which is typical in the sense of Baire category, shows a simple, but highly non-intuitive curvature behaviour: at almost all of its boundary points, in the sense of measure, all curvatures are zero, but there is also a dense and uncountable set of boundary points at which all curvatures are infinite. The purpose of this paper is to find a counterpart to this phenomenon for typical convex bodies of given constant width. Such bodies cannot have zero curvatures. A main result says that for a typical $n$-dimensional convex body of constant width $1$ (without loss of generality), at almost all boundary points, in the sense of measure, all curvatures are equal to $1$. (In contrast, note that a ball of width $1$ has radius $1/2$, hence all its curvatures are equal to $2$.) Since the property of constant width is linear with respect to Minkowski addition, the proof requires recourse to a linear curvature notion, which is provided by the tangential radii of curvature.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-04-28T15:24:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52A20",
      "53A07"
    ],
    "keywords": [
      "constant width",
      "typical curvature behaviour",
      "boundary points",
      "dimensional convex body",
      "linear curvature notion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1404.7019B"
    }
  }
}