### Abstract

Given a region R in a Euclidean space and a distinguished point p â̂̂ R, the forbidden zone, F(R,p), is the union of all open balls with center in R having p as a common boundary point. The notion of forbidden zone, defined in [2], was shown to be instrumental in the characterization of mollified zone diagrams, a relaxation of zone diagrams, introduced by Asano, et al. [3], itself a variation of Voronoi diagrams. For a polygon P, we derive formulas for the area and circumference of F(P,p) when p is fixed, and for minimum areas and circumferences when p ranges in P. These optimizations associate interesting new centers to P, even when a triangle. We give some extensions to polytopes and bounded convex sets. We generalize forbidden zones by allowing p to be replaced by an arbitrary subset, with attention to the case of finite sets. The corresponding optimization problems, even for two-point sites, and their characterizations result in many new and challenging open problems.

Original language | English (US) |
---|---|

Title of host publication | Transactions on Computational Science XX |

Subtitle of host publication | Special Issue on Voronoi Diagrams and Their Applications |

Pages | 112-137 |

Number of pages | 26 |

DOIs | |

State | Published - Dec 1 2013 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
---|---|

Volume | 8110 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Transactions on Computational Science XX: Special Issue on Voronoi Diagrams and Their Applications*(pp. 112-137). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 8110). https://doi.org/10.1007/978-3-642-41905-8-8

}

*Transactions on Computational Science XX: Special Issue on Voronoi Diagrams and Their Applications.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 8110, pp. 112-137. https://doi.org/10.1007/978-3-642-41905-8-8

**On properties of forbidden zones of polygons and polytopes.** / Berkowitz, Ross; Kalantari, Bahman; Kalantari, Iraj; Menendez, David.

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

TY - GEN

T1 - On properties of forbidden zones of polygons and polytopes

AU - Berkowitz, Ross

AU - Kalantari, Bahman

AU - Kalantari, Iraj

AU - Menendez, David

PY - 2013/12/1

Y1 - 2013/12/1

N2 - Given a region R in a Euclidean space and a distinguished point p â̂̂ R, the forbidden zone, F(R,p), is the union of all open balls with center in R having p as a common boundary point. The notion of forbidden zone, defined in [2], was shown to be instrumental in the characterization of mollified zone diagrams, a relaxation of zone diagrams, introduced by Asano, et al. [3], itself a variation of Voronoi diagrams. For a polygon P, we derive formulas for the area and circumference of F(P,p) when p is fixed, and for minimum areas and circumferences when p ranges in P. These optimizations associate interesting new centers to P, even when a triangle. We give some extensions to polytopes and bounded convex sets. We generalize forbidden zones by allowing p to be replaced by an arbitrary subset, with attention to the case of finite sets. The corresponding optimization problems, even for two-point sites, and their characterizations result in many new and challenging open problems.

AB - Given a region R in a Euclidean space and a distinguished point p â̂̂ R, the forbidden zone, F(R,p), is the union of all open balls with center in R having p as a common boundary point. The notion of forbidden zone, defined in [2], was shown to be instrumental in the characterization of mollified zone diagrams, a relaxation of zone diagrams, introduced by Asano, et al. [3], itself a variation of Voronoi diagrams. For a polygon P, we derive formulas for the area and circumference of F(P,p) when p is fixed, and for minimum areas and circumferences when p ranges in P. These optimizations associate interesting new centers to P, even when a triangle. We give some extensions to polytopes and bounded convex sets. We generalize forbidden zones by allowing p to be replaced by an arbitrary subset, with attention to the case of finite sets. The corresponding optimization problems, even for two-point sites, and their characterizations result in many new and challenging open problems.

UR - http://www.scopus.com/inward/record.url?scp=84892705582&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84892705582&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-41905-8-8

DO - 10.1007/978-3-642-41905-8-8

M3 - Conference contribution

SN - 9783642419041

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 112

EP - 137

BT - Transactions on Computational Science XX

ER -