  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.
