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 A of the space A of allowable angles ("The continuity conjecture"), and furthermore, V n (a ∈ ∂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][338][339][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 H 3 , and sharp estimates on the diameter of a polyhedron in terms of the length of the shortest polar geodesic.

Two-sided combinatorial volume bounds for non-obtuse hyperbolic polyhedra

“…A generalization of Milnor's continuity conjecture by Rivin implies that V is continuous on Ω(P) [20]. Hence by Schläfli's formula, the deformations in Corollaries 5 and 6 are volume nonincreasing.…”

Combinatorial Ricci flows and the hyperbolization of a class of compact 3-manifolds

“…Some other properties of this volume function can be found in [Riv08,Sch02]. Let L denote the set of vectors (l 12 , • • • , l 34 ) ⊂ R 6 >0 such that there exists a hyper-ideal tetrahedron having l ij as the length of the edge e ij for any {i, j} ⊂ {1, 2, 3, 4}.…”

Volume and rigidity of hyperbolic polyhedral 3‐manifolds

“…Proposition The volume function has the following properties. (a) The volume function is smooth and has positive definite Hessian matrix at each point in . (b) The function can be extended continuously to the compact closure of in , where …”

“…By the work of Schlenker , is smooth and strictly convex on . A theorem of Rivin shows that can be extended continuously to the compact closure of . Definition We call a map a generalized hyper‐ideal metric on .…”

“…Recall that is the space of dihedral angle vectors of hyper‐ideal tetrahedra defined by and let be its closure, that is, By , the volume function on is continuous and convex. To study the volume optimization, we need to classify points in .…”