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

摘要

We describe a characterization of convex polyhedra in $\h^3$ in terms oftheir dihedral angles, developed by Rivin. We also describe some geometric andcombinatorial consequences of that theory. One of these consequences is acombinatorial characterization of convex polyhedra in $\E^3$ all of whosevertices lie on the unit sphere. That resolves a problem posed by Jakob Steinerin 1832.