An obstruction to Delaunay triangulations in Riemannian manifolds

TL;DR

This paper investigates the limitations of Delaunay triangulations in Riemannian manifolds, demonstrating that increased sample density alone does not guarantee a triangulation in dimensions higher than two.

Contribution

It reveals that stronger conditions than density are necessary for Delaunay complexes to triangulate manifolds in higher dimensions, challenging previous assumptions.

Findings

Sample density alone is insufficient for triangulation in dimensions > 2

Delaunay complexes may fail to triangulate higher-dimensional manifolds

Results clarify limitations of Delaunay methods in Riemannian geometry

Abstract

Delaunay has shown that the Delaunay complex of a finite set of points $P$ of Euclidean space $\mathbb{R}^m$ 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.