Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6416027 | Linear Algebra and its Applications | 2016 | 24 Pages |
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 Ïs, associates to each set Tâs 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 Fâs, the family of sets Tâs such that Ïs(T)=F. More in detail, the paper analyzes relationships between the elements of this family, Ïsâ1(F), and the linear representations of F, provides explicit formulas for maximal and minimal elements of Ïsâ1(F), and studies the closure operator that assigns, to each closed set T containing s, the largest element of Ïsâ1(F), where F=VT(s).