On the Existence of a Neutral Region

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. However, in general this is not true, as simple counterexamples show, but we present a simple necessary and sufficient condition ensuring the non-existence of a neutral region. We discuss a similar phenomenon regarding 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. It was claimed in several works that some of these objects induce a decomposition in which a neutral region must exist, but no proof has been given to this claim. We show that this assertion is true in a wide class of cases, but not in general.