Continuity of Volumes – on a Generalization of a Conjecture of J.w.milnor

A. In his paper [6] 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 prove of this conjecture was recently given by F. Luo ([5]). In this paper we give a simple proof of this conjecture, provemuch 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.