The Voronoi inverse mapping

Given an arbitrary set T in the Euclidean space whose elements are called sites, and a particular site s, the Voronoi cell of s, denoted by VT(s), consists of all points closer to s than to any other site. The Voronoi mapping of s, denoted by ψs, associates to each set T∋s the Voronoi cell VT(s) of s w.r.t. T. These Voronoi cells are solution sets of linear inequality systems, so they are closed convex sets. In this paper we study the Voronoi inverse problem consisting in computing, for a given closed convex set F∋s, the family of sets T∋s such that ψs(T)=F. More in detail, the paper analyzes relationships between the elements of this family, ψs−1(F), and the linear representations of F, provides explicit formulas for maximal and minimal elements of ψs−1(F), and studies the closure operator that assigns, to each closed set T containing s, the largest element of ψs−1(F), where F=VT(s).