Weighted Voronoi-Delaunay dual on polyhedral surfaces and its finiteness

We aim to give a strict proof of the existence and uniqueness of the weighted Voronoi decomposition and the dual weighted Delaunay triangulation on Euclidean and hyperbolic polyhedral surface as well as hyperbolic surface with geodesic boundaries. Since the former definition of the Voronoi cell may not be simply connected, we slightly adjust the definition. Our proof is to construct an isotopic map instead of using the edge-flipping algorithm, which is a generalization of the one by Dyer et al. The main theorem of this paper is a lemma for proving the existence of the inversive distance circle packing.