Given a region R in a Euclidean space and a distinguished point p â̂̂ 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 , was 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. 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.