TY - GEN

T1 - Distance k-sectors exist

AU - Imai, Keiko

AU - Kawamura, Akitoshi

AU - Matoušek, Jiří

AU - Reem, Daniel

AU - Tokuyama, Takeshi

PY - 2010/7/30

Y1 - 2010/7/30

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 C i-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 C i-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

UR - http://www.scopus.com/inward/record.url?scp=77954660343&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77954660343&partnerID=8YFLogxK

U2 - 10.1145/1810959.1810996

DO - 10.1145/1810959.1810996

M3 - Conference contribution

AN - SCOPUS:77954660343

SN - 9781450300162

T3 - Proceedings of the Annual Symposium on Computational Geometry

SP - 210

EP - 215

BT - Proceedings of the 26th Annual Symposium on Computational Geometry, SCG'10

T2 - 26th Annual Symposium on Computational Geometry, SoCG 2010

Y2 - 13 June 2010 through 16 June 2010

ER -