  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.
