### 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 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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Title of host publication | Proceedings of the 26th Annual Symposium on Computational Geometry, SCG'10 |

Pages | 216-221 |

Number of pages | 6 |

DOIs | |

Publication status | Published - Jul 30 2010 |

Externally published | Yes |

Event | 26th Annual Symposium on Computational Geometry, SoCG 2010 - Snowbird, UT, United States Duration: Jun 13 2010 → Jun 16 2010 |

### Publication series

Name | Proceedings of the Annual Symposium on Computational Geometry |
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### Other

Other | 26th Annual Symposium on Computational Geometry, SoCG 2010 |
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Country | United States |

City | Snowbird, UT |

Period | 6/13/10 → 6/16/10 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Geometry and Topology
- Computational Mathematics

### Cite this

*Proceedings of the 26th Annual Symposium on Computational Geometry, SCG'10*(pp. 216-221). (Proceedings of the Annual Symposium on Computational Geometry). https://doi.org/10.1145/1810959.1810997

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

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the 26th Annual Symposium on Computational Geometry, SCG'10.*Proceedings of the Annual Symposium on Computational Geometry, pp. 216-221, 26th Annual Symposium on Computational Geometry, SoCG 2010, Snowbird, UT, United States, 6/13/10. https://doi.org/10.1145/1810959.1810997

}

TY - GEN

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

AU - Kawamura, Akitoshi

AU - Matoušek, Jiřŕ

AU - Tokuyama, Takeshi

PY - 2010/7/30

Y1 - 2010/7/30

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 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 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=77954656727&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77954656727&partnerID=8YFLogxK

U2 - 10.1145/1810959.1810997

DO - 10.1145/1810959.1810997

M3 - Conference contribution

SN - 9781450300162

T3 - Proceedings of the Annual Symposium on Computational Geometry

SP - 216

EP - 221

BT - Proceedings of the 26th Annual Symposium on Computational Geometry, SCG'10

ER -