## Zone and double zone diagrams in abstract spaces

### Volume 115 / 2009

#### Abstract

A zone diagram of order $n$ is a relatively new concept which was first defined and studied by T. Asano, J. Matou\v{s}ek and T. Tokuyama. It can be interpreted as a state of equilibrium between $n$ mutually hostile kingdoms. Formally, it is a fixed point of a certain mapping. These authors considered the Euclidean plane with finitely many singleton-sites 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 with 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.