A characterization of convex hyperbolic polyhedra and of convex polyhedra inscribed in the sphere

Abstract

We describe a characterization of convex polyhedra in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H cubed"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{H^3}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in terms of their dihedral angles, developed by Rivin. We also describe some geometric and combinatorial consequences of that theory. One of these consequences is a combinatorial characterization of convex polyhedra in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E cubed"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>E</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{E^3}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> all of whose vertices lie on the unit sphere. That resolves a problem posed by Jakob Steiner in 1832.