Distance Trisector Curves in Regular Convex Distance Me
ISVD '06 Proceedings of the 3rd International Symposium on Voronoi Diagrams in Science and Engineering
Distance Trisector of Segments and Zone Diagram of Segments in a Plane
ISVD '07 Proceedings of the 4th International Symposium on Voronoi Diagrams in Science and Engineering
Zone Diagrams: Existence, Uniqueness, and Algorithmic Challenge
SIAM Journal on Computing
An Algorithm for Computing Voronoi Diagrams of General Generators in General Normed Spaces
ISVD '09 Proceedings of the 2009 Sixth International Symposium on Voronoi Diagrams
Zone diagrams in Euclidean spaces and in other normed spaces
Proceedings of the twenty-sixth annual symposium on Computational geometry
Computational Geometry: Theory and Applications
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The bisector of two nonempty sets P and Q in a metric space is the set of all points with equal distance to P and to Q. A distance k-sector of P and Q, where k ≥ 2 is an integer, is a (k-1)-tuple (C1, C2, ..., Ck-1) such that Ci is the bisector of Ci-1 and Ci+1 for every i= 1, 2, ..., k-1, where C0 = P and Ck = Q. This notion, for the case where P and Q are points in Euclidean plane, was introduced by Asano, Matousek, and Tokuyama, motivated by a question of Murata in VLSI design. They established the existence and uniqueness of the distance trisector in this special case. We prove the existence of a distance k-sector for all k and for every two disjoint, nonempty, closed sets P and Q in Euclidean spaces of any (finite) dimension, or more generally, in proper geodesic spaces (uniqueness remains open). The core of the proof is a new notion of k-gradation for P and Q, whose existence (even in an arbitrary metric space) is proved using the Knaster-Tarski fixed point theorem, by a method introduced by Reem and Reich for a slightly different purpose.