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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  and shown to be instrumental in the characterization of mollified zone diagrams, a relaxation of zone diagrams, introduced by Asano, et al. , 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.