TY - JOUR

T1 - Distance k-sectors exist

AU - Imai, Keiko

AU - Kawamura, Akitoshi

AU - Matoušek, Jiří

AU - Reem, Daniel

AU - Tokuyama, Takeshi

N1 - Funding Information:
A.K. is supported by the Nakajima Foundation and the Natural Sciences and Engineering Research Council of Canada. The part of this research by T.T. was partially supported by the JSPS Grant-in-Aid for Scientific Research (B) 18300001.

PY - 2010/11

Y1 - 2010/11

N2 - The bisector of two nonempty sets P and Q in Rd 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 R2, was introduced by Asano, Matoušek, and Tokuyama, motivated by a question of Murata in VLSI design. They established the existence and uniqueness of the distance 3-sector 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 (uniqueness remains open), or more generally, in proper geodesic spaces. 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.

AB - The bisector of two nonempty sets P and Q in Rd 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 R2, was introduced by Asano, Matoušek, and Tokuyama, motivated by a question of Murata in VLSI design. They established the existence and uniqueness of the distance 3-sector 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 (uniqueness remains open), or more generally, in proper geodesic spaces. 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.

KW - Distance k-sectors

KW - Knaster-Tarski fixed point theorem

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U2 - 10.1016/j.comgeo.2010.05.001

DO - 10.1016/j.comgeo.2010.05.001

M3 - Article

AN - SCOPUS:77954657071

VL - 43

SP - 713

EP - 720

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

SN - 0925-7721

IS - 9

ER -