On Bisectors for Convex Distance Functions

It is well known that the construction of Voronoi diagrams is based on the notion of bisector of two given points. Already in normed linear spaces, bisectors have a complicated structure and can, for many classes of norms, only be described with the help of topological methods. Even more general, we present results on bisectors for convex distance functions (gauges). Let C, with the origin o from its interior, be the compact, convex set inducing a convex distance function (gauge) in the plane, and let B(-x,x) be the bisector of - x and x, i.e., the set of points z such that the distance (measured with the convex distance function induced by C) from z to - x equals that from z to x. For example, we prove the following characterization of the Euclidean norm within the family of all convex distance functions: if the set L of points x in the boundary ∂C of C that creates B(-x, x) as a straight line has non-empty interior with respect to ∂C, then C is an ellipse centered at the origin. For the subcase of normed planes we give an easier approach, extending the result also to higher dimensions.