Distance k-Sectors Exist

TL;DR

This paper proves the existence of distance k-sectors for any two disjoint, nonempty, closed sets in Euclidean and proper geodesic spaces, extending previous results limited to Euclidean planes.

Contribution

It generalizes the existence of distance k-sectors from Euclidean planes to higher-dimensional Euclidean and proper geodesic spaces, introducing a new concept of k-gradation.

Findings

Existence of distance k-sectors for all k in Euclidean spaces of any dimension.

Introduction of k-gradation concept using fixed point theorem.

Extension of previous Euclidean plane results to more general metric spaces.

Abstract

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 is an integer, is a (k-1)-tuple (C_1, C_2, ..., C_{k-1}) such that C_i is the bisector of C_{i-1} and C_{i+1} for every i = 1, 2, ..., k-1, where C_0 = P and C_k = 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.