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Journal for Geometry and Graphics 7 (2003), No. 1, 041--052
Copyright Heldermann Verlag 2003

A Voronoi Poset

Roderik C. Lindenbergh
Mathematical Geodesy and Positioning, Delft University of Technology, Kluyverweg 1, 2629 HS Delft, The Netherlands,

Given a set S of n points in general position, we consider all k-th order Voronoi diagrams on S, for k = 1,...,n, simultaneously. We recall symmetry relations for the number of cells, number of vertices and number of circles of certain orders. We introduce a poset Π(S) that consists of the k-th order Voronoi cells for all k = 1,...,n, that occur for some set S. We prove that there exists a rank function on Π(S) and moreover that the number of elements of odd rank equals the number of elements of even rank of Π(S), provided that n is odd.

Keywords: k-th order Voronoi diagrams, k-sets, posets, point configurations.

MSC: 52B55, 68U05.

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