## Abstract

A zone diagram is a new variation of the classical notion of Voronoi diagram. Given points (sites) p_{1}, ⋯;, p_{n} in the plane, each p_{i}, is assigned a region R_{i}, but in contrast to the ordinary Voronoi diagrams, the union of the R_{i} has a nonempty complement, the neutral zone. The defining property is that each R_{i} consists of all x ∈ ℝ^{2} that lie closer (non-strictly) to p_{i} than to the union of all the other R_{j}, j ≠ i. Thus, the zone diagram is defined implicitly, by a "fixed-point property," and neither its existence nor its uniqueness seem obvious. We establish existence using a general fixed-point result (a consequence of Schauder's theorem or Kakutani's theorem); this proof should generalize easily to related settings, say higher dimensions. Then we prove uniqueness of the zone diagram, as well as convergence of a natural iterative algorithm for computing it, by a geometric argument, which also relies on a result for the case of two sites in an earlier paper. Many challenging questions remain open.

Original language | English |
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Title of host publication | Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2007 |

Publisher | Association for Computing Machinery |

Pages | 756-765 |

Number of pages | 10 |

Volume | 07-09-January-2007 |

ISBN (Electronic) | 9780898716245 |

Publication status | Published - 2007 Jan 1 |

Event | 18th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2007 - New Orleans, United States Duration: 2007 Jan 7 → 2007 Jan 9 |

### Other

Other | 18th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2007 |
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Country | United States |

City | New Orleans |

Period | 07/1/7 → 07/1/9 |

## ASJC Scopus subject areas

- Software
- Mathematics(all)