An Obstruction to Delaunay Triangulations in Riemannian Manifolds

Abstract

Delaunay has shown that the Delaunay complex of a finite set of points $$P$$ of Euclidean space $$\mathbb {R}^m$$ 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.