On farthest Voronoi cells

© 2019 Elsevier Inc. Given an arbitrary set T in the Euclidean space Rn, whose elements are called sites, and a particular site s, the farthest Voronoi cell of s, denoted by FT(s), consists of all points which are farther from s than from any other site. In this paper we study farthest Voronoi cells and diagrams corresponding to arbitrary (possibly infinite) sets. More in particular, we characterize, for a given arbitrary set T, those s∈T such that FT(s) is nonempty and study the geometrical properties of FT(s) in that case. We also characterize those sets T whose farthest Voronoi diagrams are tesselations of the Euclidean space, and those sets that can be written as FT(s) for some T⊂Rn and some s∈T.