Constant Minkowskian width in terms of double normals

Abstract

We extend the notion of a double normal of a convex body from smooth, strictly convex Minkowski planes to arbitrary two-dimensional real, normed, linear spaces in two different ways. Then, for both of these ways, we obtain the following characterization theorem: a convex body K in a Minkowski plane is of constant Minkowskian width iff every chord I of K splits K into two compact convex sets K1 and K2 such that I is a Minkowskian double normal of K1 or K2. Furthermore, the Euclidean version of this theorem yields a new characterization of d-dimensional Euclidean ball where d ≥ 3.