2005/12/02 by Igor Rivin, Rivin, Igor
Mathematics · #52A15 #53C23 #53C45 #FOS: Mathematics #Geometric Topology (math.GT) #Metric Geometry (math.MG) #math.GT #math.MG #msc:52A15 #msc:53C23 #msc:53C45
paper · pdf · doi:10.48550/arxiv.math/0512065
supercedes math.GT/0502543
arxiv created 2005/12/02 · arxiv updated 2009/12/01
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.