On Properties of Forbidden Zones of Polygons and Polytopes

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. For a polytope, the forbidden zone is the union of open balls centered at its vertices. The notion of forbidden zone was defined in [1] and shown to be instrumental in the characterization of mollified zone diagrams, a relaxation of zone diagrams, introduced by Asano, et al. [2], itself a variation of Voronoi diagrams. In this article we focus on properties of F(P, p) where P is a convex polygon. We derive formulas for the area and circumference of F(P, p) when p is fixed, and for minimum areas and circumferences when p is allowed to range in P. Moreover, we give formulas for the area and circumference of a flower-shaped region corresponding to intersecting circles in F(P, p), and for optimal values as p ranges in P. We also extend our formulas for p ∈ P. We then develop a formula for the area of the intersection of circles having a common boundary point. The optimization problems associate interesting centers to a polygon, even to a triangle, different from their classical versions. Aside from geometric interest, applications could exist. Finally, we extend some of the above results and optimizations to arbitrary polytopes and bounded convex sets.