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.
This paper is an extended version of [1] and is dedicated to the memory of Sergio de Biasi.
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Berkowitz, R., Kalantari, B., Kalantari, I., Menendez, D. (2013). On Properties of Forbidden Zones of Polygons and Polytopes. In: Gavrilova, M.L., Tan, C.J.K., Kalantari, B. (eds) Transactions on Computational Science XX. Lecture Notes in Computer Science, vol 8110. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41905-8_8
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DOI: https://doi.org/10.1007/978-3-642-41905-8_8
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