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. (2000). 46B04, 46B20, 51F15, 52A21. Mathematics Subject Classification

Monge Points, Euler Lines, and Feuerbach Spheres in Minkowski Spaces

“…And of course it is an interesting task for geometers to generalize notions like orthogonality (see [1,2,5]), orthocentricity (cf. [8,34,40,45]), isometries (see [36,38]), and regularity (see [37]) in absence of an inner product. In case of regularity, we may ask which figures are special, and what are useful concepts to describe their degree of symmetry in normed planes and spaces?…”

Isometries of Minkowski geometries

“…The scarcity of isometries of non-Euclidean Minkowski planes motivated H. Martini and M. Spirova to introduce left reflections as a conceptual tool for investigating strictly convex Minkowski planes (see [20]). In general, left reflections are not isometries, but they are strongly related to the notion of Birkhoff orthogonality: if every left reflection in a strictly convex Minkowski plane preserves Birkhoff orthogonality, then this plane is a Radon plane.…”

Circle configurations in strictly convex normed planes

“…Using that in a flat Möbius plane Apollonius' problem can be solved in general (see Groh [7]), we transfer Groh's results to an arbitrary strictly convex, smooth normed plane. For related properties of circles in strictly convex Minkowski planes see [1], [10], and [11].…”