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 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. 1 Delaunay complex and Delaunay triangulation Let (M, dM) be a metric space, and let P be a finite set of points of M. An empty ball is an open ball in the metric dM that contains no point from P . We say that an empty ball B is maximal if no other empty ball with the same centre properly contains B. A Delaunay ball is a maximal empty ball. A simplex σ is a Delaunay simplex if there exists some Delaunay ball B that circumscribes σ, i.e., such that the vertices of σ belong to ∂B ∩ P . The Delaunay complex is the set of Delaunay simplices, and is denoted DelM(P ). It is an abstract simplicial complex and so defines a topological space, |DelM(P )|, called its carrier . We say that DelM(P ) triangulates M if |DelM(P )| is homeomorphic to M. A Delaunay triangulation of M is a homeomorphism H : |DelM(P )| → M. The Voronoi cell associated with p ∈ P is given by VM(p) = {x ∈M | dM(x, p) ≤ dM(x, q) for all q ∈ P}. More generally, a Voronoi face is the intersection of a set of Voronoi cells: given σ = {p0, . . . , pk} ⊂ P , we define the associated Voronoi face as