On farthest Voronoi cells

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 corresp...

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Bibliographic Details
Published in:Linear algebra and its applications 2019-12, Vol.583, p.306-322
Main Authors: Goberna, M.A., Martínez-Legaz, J.E., Todorov, M.I.
Format: Article
Language:English
Subjects:
Boundedly exposed points
Euclidean geometry
Euclidean space
Farthest Voronoi cells
Linear algebra
Linear inequality systems
Voronoi graphs
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Summary: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.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2019.09.002