### Abstract

A zone diagram is a new variation of the classical notion of the 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 Ri has a nonempty complement, the neutral zone. The defining property is that each R_{i} consists of all x ε ℝ^{2} that lie closer (nonstrictly) 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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Pages (from-to) | 1182-1198 |

Number of pages | 17 |

Journal | SIAM Journal on Computing |

Volume | 37 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2007 Dec 1 |

### Keywords

- Computational geometry
- Distance trisector curve
- Voronoi diagram
- Zone diagram

### ASJC Scopus subject areas

- Computer Science(all)
- Mathematics(all)

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## Cite this

*SIAM Journal on Computing*,

*37*(4), 1182-1198. https://doi.org/10.1137/06067095X