An Obstruction to Delaunay Triangulations in Riemannian Manifolds

Delaunay has shown that the Delaunay complex of a finite set of points P of Euclidean space ℝ^m triangulates the convex hull of P, provided that P 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 P are required. A natural one is to assume that P 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.