An Obstruction to Delaunay Triangulations in Riemannian Manifolds

Delaunay has shown that the Delaunay complex of a finite set of points of Euclidean space triangulates the convex hull of provided that satisfies a mild genericity property. Voronoi diagrams and Delaunay complexes can be defined for arbitrary Riemannian manifolds. However, Delaunay's genericity assumption no longer guarantees that the Delaunay complex will yield a triangulation; stronger assumptions on are required. A natural one is to assume that is sufficiently dense. Although results in this direction have been claimed, we show that sample density alone is insufficient to ensure that the Delaunay complex triangulates a manifold of dimension greater than 2.