Volumes and degeneration -- on a conjecture of J. W. Milnor
| dc.creator | Rivin, Igor | |
| dc.date | 2005-12-02 | |
| dc.date.accessioned | 2026-07-07T06:54:49Z | |
| dc.date.available | 2026-07-07T06:54:49Z | |
| 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 (``The continuity conjecture''), and furthermore, the limit at a boundary point is equal to 0 if and only if the point lies in the closure of the space of angles of Euclidean tetrahedra (``the Vanishing Conjecture''). A proof of the Continuity Conjecture was given by F. Luo -- Luo's argument uses Kneser's formula for the volume together with some delicate geometric estimates). In this paper we give a simple proof of both parts of Milnor's conjecture, prove much sharper regularity results, and then extend the method to apply to all convex polytopes. We also give a precise description of the boundary of the space of angles of convex polyhedra in and sharp estimates on the diameter of a polyhedron in terms of the length of the shortest closed geodesic of the polar metric. | |
| dc.description | supercedes math.GT/0502543 | |
| dc.identifier | https://arxiv.org/abs/math/0512065 | |
| dc.identifier | http://arxiv.org/abs/math/0512065 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/106074 | |
| dc.subject | Geometric Topology | |
| dc.subject | Metric Geometry | |
| dc.subject | 52A15; 53C23; 53C45 | |
| dc.title | Volumes and degeneration -- on a conjecture of J. W. Milnor | |
| dc.type | text |