On the existence of a neutral region

TL;DR

This paper investigates conditions under which neutral regions can or cannot exist in Voronoi and related diagrams, providing necessary and sufficient criteria and clarifying misconceptions about their properties.

Contribution

It establishes a simple necessary and sufficient condition for the non-existence of neutral regions in Voronoi and similar diagrams, clarifying prior claims and extending understanding.

Findings

Neutral regions do not always exist in Voronoi decompositions.

A simple criterion determines when neutral regions are absent.

The paper clarifies misconceptions about neutral regions in zone diagrams.

Abstract

Consider a given space, e.g., the Euclidean plane, and its decomposition into Voronoi regions induced by given sites. It seems intuitively clear that each point in the space belongs to at least one of the regions, i.e., no neutral region can exist. As simple counterexamples show this is not true in general, but we present a simple necessary and sufficient condition ensuring the non-existence of a neutral region. We discuss a similar phenomenon concerning recent variations of Voronoi diagrams called zone diagrams, double zone diagrams, and (double) territory diagrams. These objects are defined in a somewhat implicit way and they also induce a decomposition of the space into regions. In several works it was claimed without providing a proof that some of these objects induce a decomposition in which a neutral region must exist. We show that this assertion is true in a wide class of cases but not in general. We also discuss other properties related to the neutral region, among them a one related to the concentration of measure phenomenon.