  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 M contains or is contained in the restricted Delaunay
triangulation of L.
