Volumes and degeneration -- on a conjecture of J. W. Milnor

Details Volumes and degeneration -- on a conjecture of J. W. Milnor Volumes and degeneration -- on a conjecture of J. W. Milnor

2005-12-02

arxiv.org/abs/math/0512065v1

Mathematics

Geometric Topology

supercedes math.GT/0502543

Scientific paper

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.

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