Volumes of degenerating polyhedra — on a conjecture of J. W. Milnor

Abstract

In his paper (Milnor, The Schläfli Differential equality, Collected Works, vol 1, Publish or Perish, Houston 1994) Milnor conjectured that the volume V _n of compact n-dimensional hyperbolic and spherical simplices, as a function of the dihedral angles, extends continuously to the closure ${\overline{\mathbb{A}}}$ of the space ${\mathbb{A}}$ of allowable angles (“The continuity conjecture”), and furthermore, ${V_n(a\in\partial \mathbb{A}) = 0}$ if and only if a lies in the closure of the space of angles of Euclidean simplices (“the Vanishing Conjecture”). A proof of the Continuity Conjecture was given by Luo (Commun. Contemp. Math. 8(3), 411–431, 2006—Luo’s argument uses Kneser’s formula, Deutsche Mathematik 1, 337–340, 1936 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 ${\mathbb{H}^3}$ , and sharp estimates on the diameter of a polyhedron in terms of the length of the shortest polar geodesic.