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Delaunay has shown that the Delaunay complex of a finite set of points $P$ ofEuclidean space $\mathbb{R}^m$ triangulates the convex hull of $P$, providedthat $P$ satisfies a mild genericity property. Voronoi diagrams and Delaunaycomplexes can be defined for arbitrary Riemannian manifolds. However,Delaunay's genericity assumption no longer guarantees that the Delaunay complexwill yield a triangulation; stronger assumptions on $P$ are required. A naturalone is to assume that $P$ is sufficiently dense. Although results in thisdirection have been claimed, we show that sample density alone is insufficientto ensure that the Delaunay complex triangulates a manifold of dimensiongreater than 2.

An obstruction to Delaunay triangulations in Riemannian manifolds