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Continuity of volumes -- on a generalization of a conjecture of J. W. Milnor

Igor Rivin
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Abstract

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.

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Igor Rivin. 2005-03-22. Continuity of volumes -- on a generalization of a conjecture of J. W. Milnor. https://arxiv.org/abs/math/0502543

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