### Abstract

Zone diagrams are a variation on the classical concept of Voronoi diagrams. 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, Matoušek, and Tokuyama proved the existence and uniqueness of a zone diagram for point sites in the 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.

Original language | English |
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Pages (from-to) | 1201-1221 |

Number of pages | 21 |

Journal | Mathematische Annalen |

Volume | 354 |

Issue number | 4 |

DOIs | |

Publication status | Published - Nov 2012 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

*Mathematische Annalen*,

*354*(4), 1201-1221. https://doi.org/10.1007/s00208-011-0761-1