Zone and double zone diagrams in abstract spaces

TL;DR

This paper generalizes the concept of zone diagrams from Euclidean spaces to abstract m-spaces, providing new existence, non-uniqueness results, explicit examples, and a combinatorial game interpretation with an algorithm.

Contribution

It extends zone diagram theory to m-spaces, introduces order-theoretic proofs, and explores new phenomena and computational aspects.

Findings

Existence and non-uniqueness results in m-spaces

Explicit examples illustrating new phenomena

Algorithm for finding stable configurations in a combinatorial game

Abstract

A zone diagram is a relatively new concept which was first defined and studied by T. Asano, J. Matousek and T. Tokuyama. It can be interpreted as a state of equilibrium between several mutually hostile kingdoms. Formally, it is a fixed point of a certain mapping. These authors considered the Euclidean plane and proved the existence and uniqueness of zone diagrams there. In the present paper we generalize this concept in various ways. We consider general sites in m-spaces (a simple generalization of metric spaces) and prove several existence and (non)uniqueness results in this setting. In contrast to previous works, our (rather simple) proofs are based on purely order theoretic arguments. Many explicit examples are given, and some of them illustrate new phenomena which occur in the general case. We also re-interpret zone diagrams as a stable configuration in a certain combinatorial game, and provide an algorithm for finding this configuration in a particular case.