On the Maximum Apollonian Sets

Abstract

In this study, the Apollonian sets are analytically examined in the plane equipped with the maximum distance function. It is determined that the maximum Apollonian set is not a maximum circle, but the closed simple rectilinear figure. Also, the classifications of the maximum Apollonian sets are given. In the case that the pair of points is on the gradual line or the steep line, the maximum Apollonian set is classified with respect to the condition.