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arXiv · 0803.1296

On the Topology of the Restricted Delaunay Triangulation and Witness Complex in Higher Dimensions

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

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Steve Y. Oudot. 2008-03-09. On the Topology of the Restricted Delaunay Triangulation and Witness Complex in Higher Dimensions. https://arxiv.org/abs/0803.1296

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