A non-uniqueness theorem in the theory of Voronoi sets (Q1187092)

If \(\{D_ i\}_{0\leq i\leq n}\) is a finite collection of non-empty, bounded, open and simply connected subsets of \(\mathbb{R}^ 2\) which satisfy \(D_ i\varsubsetneq D_ 0\), \(1\leq i\leq n\), \(\bar D_ i\cap \bar D_ j=\emptyset\) for \(i\neq j\), and if \(\Omega=D_ 0\setminus\bigcup^ n_{i=1}\bar D_ i\), then the Voronoi diagram of \(\Omega\), \(\text{Vor}(\Omega)\) is defined to be \(\{(x,y)\in \Omega\mid \text{Near}(x,y)\) contains more than one point\}, where \(\text{Near}(x,y)\) is the set of points in \(\partial \Omega\) closest to \((x,y)\) (the metric being Euclidean). The author gives an example of a domain \(D_ 0\) and two different collections of subsets \(\{D_ i\}_{1\leq i\leq 2}\) and \(\{D_ i'\}_{1\leq i\leq 2}\), which have the same Voronoi diagram, i.e. \(\text{Vor}(\Omega)=\text{Vor}(\Omega')\), for \(\Omega=D_ 0\setminus\bigcup^ 2_{i=1} D_ i\) and \(\Omega'=D_ 0\setminus \bigcup^ 2_{i=1}\bar D_ i'\).