An Obstruction to Delaunay Triangulations in Riemannian Manifolds

Abstract

Delaunay has shown that the Delaunay complex of a finite set of points P of Euclidean space 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. Delaunay complex and Delaunay triangulationLet (M, d M ) be a metric space, and let P be a finite set of points of M. An empty ball is an open ball in the metric d M that contains no point from P . We say that an empty ball B is maximal if no other empty ball with the same centre properly contains B. A Delaunay ball is a maximal empty ball.A simplex σ is a Delaunay simplex if there exists some Delaunay ball B that circumscribes σ, i.e., such that the vertices of σ belong to ∂B ∩ P . The Delaunay complex is the set of Delaunay simplices, and is denoted Del M (P ). It is an abstract simplicial complex and so defines a topological space, |Del M (P )|, called its carrier . We say that DelThe Voronoi cell associated with p ∈ P is given byMore generally, a Voronoi face is the intersection of a set of Voronoi cells: given σ = {p 0 , . . . , p k } ⊂ P , we define the associated Voronoi face asIt follows that σ is a Delaunay simplex if and only if V M (σ) = ∅. In this case, every point in V M (σ) is the centre of a Delaunay ball for σ. Thus every Voronoi face corresponds to a Delaunay

Hierarchical stable election protocol for WSN‐based IoT inhabitant and environmental monitoring applications

“…The Voronoi diagram divides a region into two‐ or three‐dimensional fields that have been measured using Euclidean distance. The Delaunay triangulation, the mechanism used for creating a Voronoi diagram, refers to the nerve of the cell 44 . The use of Voronoi in node deployment ensures that the network quickly converges.…”

“…The Delaunay triangulation, the mechanism used for creating a Voronoi diagram, refers to the nerve of the cell. 44 The use of Voronoi in node deployment ensures that the network quickly converges. The following subsections discuss the node deployment scenario for 2D and 3D networks and the CH selection and cluster formation.…”

Topology and chromatic number of random $ε$-distance graphs on spheres

“…We refer the reader to [ES97; LL00; Boi+17] for all the details, and to [Vor08; Ede87; Del34] for more context. It is worth mentioning that [Boi+17] later disproved that the conditions described in [LL00] are sufficient to get geodesic Delaunay triangulations for an arbitrary Riemmannian Manifold, however, they do work for certain manifolds, including those with constant curvature, such as spheres [Boi+17, p. 1]. Definition 3.2 (Delaunay Triangulation).…”

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“…Here n will be estimated in terms of minimal number of triangles of a triangulation of M . We shall consider point configurations for which the Delaunay triangulation exists, see [1].…”