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The bisector of two nonempty sets P and Q in R^d 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 (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 R^2, 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 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.