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

Abstract

We describe a characterization of convex polyhedra in H3 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 E3 all of whose vertices lie on the unit sphere. That resolves a problem posed by Jakob Steiner in 1832. © 1992 American Mathematical Society.