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)


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.


Convex Hull Voronoi Diagram Convex Polygon Voronoi Cell Arbitrary Subset 
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© 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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