Steve Y. Oudot. On the Topology of the Restricted Delaunay Triangulation and Witness Complex in Higher Dimensions. Technical Report, Stanford University, November 2006. LANL arXiv:0803.1296v1 [cs.CG], http://arxiv.org/abs/0803.1296. Published in the full version of this paper.

Abstract:

It is a well-known fact that, under mild sampling conditions, the restricted Delaunay triangulation provides good topological approximations of 1- and 2-manifolds. We show that this is not the case for higher-dimensional manifolds, even under stronger sampling conditions. Specifically, it is not true that, for any compact closed submanifold M of R^n, and any sufficiently dense uniform sampling L of M, the Delaunay triangulation of L restricted to M is homeomorphic to M, or even homotopy equivalent to it. Besides, it is not true either that, for any sufficiently dense set W of witnesses, the witness complex of L relative to W contains or is contained in the restricted Delaunay triangulation of L.

Bibtex: 

@techreport{o-ntrdwchd-06
, author = "S. Y. Oudot"
, title = "On the topology of the restricted {Delaunay} triangulation and witness complex in higher dimensions"
, month = "November"
, year = "2006"
, institution = "Stanford University"
, note = "LANL arXiv:0803.1296v1 [cs.CG], url{http://arxiv.org/abs/0803.1296}"
}