The geometry of Minkowski spaces -- a survey. Part I

TL;DR

This survey explores fundamental geometric properties of Minkowski spaces, focusing on planar results, providing simplified proofs, and highlighting their significance in the context of finite-dimensional Banach spaces.

Contribution

It compiles elementary geometric results in Minkowski spaces, emphasizing planar cases, with simplified proofs and insights often overlooked or rediscovered in literature.

Findings

Characterization of the triangle inequality and its consequences

Geometric criteria for strict convexity and normality

Properties of equilateral triangles, circles, and polygons in Minkowski spaces

Abstract

We survey elementary results in Minkowski spaces (i.e. finite dimensional Banach spaces) that deserve to be collected together, and give simple proofs for some of them. We place special emphasis on planar results. Many of these results have often been rediscovered as lemmas to other results. In Part I we cover the following topics: The triangle inequality and consequences such as the monotonicity lemma, geometric characterizations of strict convexity, normality (Birkhoff orthogonality), conjugate diameters and Radon curves, equilateral triangles and the affine regular hexagon construction, equilateral sets, circles: intersection, circumscribed, characterizations, circumference and area, inscribed equilateral polygons.