{
  "id": "math/0502543",
  "version": "v2",
  "published": "2005-02-25T16:09:37.000Z",
  "updated": "2005-03-22T20:47:34.000Z",
  "title": "Continuity of volumes -- on a generalization of a conjecture of J. W. Milnor",
  "authors": [
    "Igor Rivin"
  ],
  "comment": "12 pages; revision has minor cosmetic changes",
  "categories": [
    "math.GT",
    "math.MG"
  ],
  "abstract": "In his paper \"On the Schlafli differential equality\", J. Milnor conjectured that the volume of n-dimensional hyperbolic and spherical simplices, as a function of the dihedral angles, extends continuously to the closure of the space of allowable angles. A proof of this has recently been given by F. Luo (see math.GT/0412208). In this paper we give a simple proof of this conjecture, prove much sharper regularity results, and then extend the method to apply to a large class of convex polytopes. The simplex argument works without change in dimensions greater than 3 (and for spherical simplices in all dimensions), so the bulk of this paper is concerned with the three-dimensional argument. The estimates relating the diameter of a polyhedron to the length of the systole of the polar polyhedron are of independent interest.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2005-03-22T20:47:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52B11",
      "52B10",
      "57M50"
    ],
    "keywords": [
      "conjecture",
      "continuity",
      "generalization",
      "schlafli differential equality",
      "spherical simplices"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......2543R"
    }
  }
}