  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.
