The Delaunay tessellation in hyperbolic space

The Delaunay tessellation of a locally finite subset of hyperbolic space is constructed using convex hulls in Euclidean space of one higher dimension. Basic properties, including the empty circumspheres condition and geometric duality with the Voronoi tessellation, are proved and compared with those of the Euclidean version. Some pathological examples are considered. Passing to the lattice-invariant case, we describe an analog of the "Epstein-Penner decomposition" of a finite-volume hyperbolic manifold M.