A Weak Characterisation of the Delaunay Triangulation

We consider a new construction, the weak Delaunay triangulation of a finite point set in a metric space, which contains as a subcomplex the traditional (strong) Delaunay triangulation. The two simplicial complexes turn out to be equal for point sets in Euclidean space, as well as in the (hemi)sphere, hyperbolic space, and certain other geometries. There are weighted and approximate versions of the weak and strong complexes in all these geometries, and we prove equality theorems in those cases also. On the other hand, for discrete metric spaces the weak and strong complexes are decidedly different. We give a short empirical demonstration that weak Delaunay complexes can lead to dramatically clean results in the problem of estimating the homology groups of a manifold represented by a finite point sample.