On Bisectors for Different Distance Functions

Let γC and γD be two convex distance functions in the plane with convex unit balls C and D. Given two points, p and q, we investigate the bisector, B(p, q), of p and q, where distance from p is measured by γC and distance from q by γD. We provide the following results. B(p, q) may consist of many connected components whose precise number can be derived from the intersection of the unit balls, C and D. The bisector can contain bounded or unbounded 2-dimensional areas. Even more surprising, pieces of the bisector may appear inside the region of all points closer to p than to q. If C and D are convex..