Reflections in strictly convex Minkowski planes

Abstract

It is known that translations, symmetries with respect to points, and the identity map are the only isometries in general (normed or) Minkowski planes. Inspired by this “difference” to the Euclidean situation, we introduce so-called left-reflections in lines for the case of strictly convex Minkowski planes, and we develop a little theory on their products, yielding also results on glide reflections. As natural consequences we obtain several new characterizations of special normed planes, such as Radon planes or the Euclidean plane. All properties of left-reflections in strictly convex normed planes derived here hold in an analogous manner for correspondingly defined right-reflections in smooth Minkowski planes.