Voronoi cells of discrete point sets

TL;DR

This paper investigates conditions under which Voronoi cells of infinite discrete point sets are polyhedral, establishing that all cells are polytopes if and only if all points are inner points, and introducing locally finitely generated sets.

Contribution

It characterizes when Voronoi cells of infinite discrete point sets are polyhedral and introduces the concept of locally finitely generated sets.

Findings

Voronoi cells are polytopes iff all points are inner points.

Locally finitely generated sets have only polyhedral Voronoi cells.

Provides a complete characterization of polyhedral Voronoi cells for infinite discrete sets.

Abstract

It is well known that all cells of the Voronoi diagram of a Delaunay set are polytopes. For a finite point set, all these cells are still polyhedra. So the question arises, if this observation holds for all discrete point sets: Are always all Voronoi cells of an arbitrary, infinite discrete point set polyhedral? In this paper, an answer to this question will be given. It will be shown that all Voronoi cells of a discrete point set are polytopes if and only if every point of the point set is an inner point. Furthermore, the term of a locally finitely generated discrete point set will be introduced and it will be shown that exactly these sets have the property of possessing only polyhedral Voronoi cells.