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On Properties of Forbidden Zones of Polygons and Polytopes

  • Ross Berkowitz
  • Bahman Kalantari
  • Iraj Kalantari
  • David Menendez
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8110)

Abstract

Given a region R in a Euclidean space and a distinguished point p ∈ 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.

Keywords

Convex Hull Voronoi Diagram Convex Polygon Voronoi Cell Arbitrary Subset 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Ross Berkowitz
    • 1
  • Bahman Kalantari
    • 1
  • Iraj Kalantari
    • 2
  • David Menendez
    • 1
  1. 1.Rutgers, the State University of New JerseyNew BrunswickUSA
  2. 2.Western Illinois UniversityMacombUSA

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