On Properties of Forbidden Zones of Polygons and Polytopes

Given a region $R$ in a Euclidean space and a distinguished point $p \in R$, the \emph{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 \cite{mollified} and shown to be instrumental in the characterization of \emph{mollified zone diagrams}, a relaxation of \emph{zone diagrams}, introduced by Asano, et al. \cite{zone diagrams}, 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 $p \not \in P$. We then develop a formula for the area of the intersection of circles having a common boundary point. The optimization problems associate interesting \emph{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 poly topes and bounded convex sets.