Zone Diagrams in Euclidean Spaces and in Other Normed Spaces

TL;DR

This paper proves the existence and uniqueness of zone diagrams for multiple sites in Euclidean and certain normed spaces, simplifying previous proofs and exploring conditions for non-uniqueness.

Contribution

It extends the existence and uniqueness results of zone diagrams to n disjoint compact sites in higher-dimensional Euclidean and normed spaces with specific properties.

Findings

Existence and uniqueness for n sites in Euclidean spaces of any finite dimension.

Existence and uniqueness for two sites in general normed spaces with smooth norms.

Counterexample showing non-uniqueness in rotund but non-smooth norms.

Abstract

Zone diagram is a variation on the classical concept of a Voronoi diagram. Given n sites in a metric space that compete for territory, the zone diagram is an equilibrium state in the competition. Formally it is defined as a fixed point of a certain "dominance" map. Asano, Matousek, and Tokuyama proved the existence and uniqueness of a zone diagram for point sites in Euclidean plane, and Reem and Reich showed existence for two arbitrary sites in an arbitrary metric space. We establish existence and uniqueness for n disjoint compact sites in a Euclidean space of arbitrary (finite) dimension, and more generally, in a finite-dimensional normed space with a smooth and rotund norm. The proof is considerably simpler than that of Asano et al. We also provide an example of non-uniqueness for a norm that is rotund but not smooth. Finally, we prove existence and uniqueness for two point sites in the plane with a smooth (but not necessarily rotund) norm.