### 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 |
---|---|

Pages (from-to) | 1201-1221 |

Number of pages | 21 |

Journal | Mathematische Annalen |

Volume | 354 |

Issue number | 4 |

DOIs | |

Publication status | Published - Nov 1 2012 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Mathematische Annalen*,

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

**Zone diagrams in Euclidean spaces and in other normed spaces.** / Kawamura, Akitoshi; Matoušek, Jiří; Tokuyama, Takeshi.

Research output: Contribution to journal › Article

*Mathematische Annalen*, vol. 354, no. 4, pp. 1201-1221. https://doi.org/10.1007/s00208-011-0761-1

}

TY - JOUR

T1 - Zone diagrams in Euclidean spaces and in other normed spaces

AU - Kawamura, Akitoshi

AU - Matoušek, Jiří

AU - Tokuyama, Takeshi

PY - 2012/11/1

Y1 - 2012/11/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84869080628&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84869080628&partnerID=8YFLogxK

U2 - 10.1007/s00208-011-0761-1

DO - 10.1007/s00208-011-0761-1

M3 - Article

AN - SCOPUS:84869080628

VL - 354

SP - 1201

EP - 1221

JO - Mathematische Annalen

JF - Mathematische Annalen

SN - 0025-5831

IS - 4

ER -