  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.
