Minimal enclosing discs, circumcircles, and circumcenters in normed planes. I. (Q419378)

Let \(\|\cdot\|\) be a norm for the vector space \({\mathbb R}^2\), and suppose we are given three non--colinear points \(t_1,t_2,t_3\) in the plane. In contrast to the Euclidean case, the intersection of two circles may contain line segments, and there may be multiple circumcircles that contain the points \(t_1,t_2,t_3\). In this article, the authors classify all possible intersections of two circles in the plane. This classification is then used to explicitly determine the regions in the plane that consist of points \(x\) for which there exists a norm such that \(x\) is the center of a circumcircle of \(t_1,t_2,t_3\). Moreover for the given norm \(\|\cdot\|\), in the case where the line segment from \(t_1\) to \(t_2\) lies in the intersection of two distinct circumcircles \({\mathcal C},{\mathcal C}'\) with centers \(c,c'\), the authors determine further restrictions on the regions where the centers can be located; in particular, it is shown that \({\mathcal C},{\mathcal C}'\) have the same radius if and only if \(c,c'\) lie on the line passing through the midpoint of \(t_1\) and \(t_3\) and the midpoint of \(t_2\) and \(t_3\).NEWLINENEWLINEThe article is largely self-contained and intended for a general mathematical audience.