Continuity of volumes -- on a generalization of a conjecture of J. W. Milnor
| dc.creator | Rivin, Igor | |
| dc.date | 2005-02-25 | |
| dc.date | 2005-03-22 | |
| dc.date.accessioned | 2026-07-07T05:17:29Z | |
| dc.date.available | 2026-07-07T05:17:29Z | |
| dc.description | 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. | |
| dc.description | 12 pages; revision has minor cosmetic changes | |
| dc.identifier | https://arxiv.org/abs/math/0502543 | |
| dc.identifier | http://arxiv.org/abs/math/0502543 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/74326 | |
| dc.subject | Geometric Topology | |
| dc.subject | Metric Geometry | |
| dc.subject | 52B11, 52B10, 57M50 | |
| dc.title | Continuity of volumes -- on a generalization of a conjecture of J. W. Milnor | |
| dc.type | text |