Distance between separating circles and points

Abstract

The family of separating circles of two finite sets S1 and S2 in the plane consists of all the circles that enclose S 1 but exclude S2. We prove that the maximum and minimum distance between a point p and any separating circle in this family can be found by examining only a finite subset of circles, although the family itself is infinite. In addition, we introduce the concept of elementary circular separations to clarify some of the properties of separating circles. © 2011 Springer-Verlag.