A complete characterization of Radon planes whose unit spheres are regular polygons

Abstract

We study the structure of the unit sphere of polygonal Radon planes from a geometric point of view. In particular, we prove that a $ 2 $-dimensional real polygonal Banach space $ \mathbb{X} $ cannot be a Radon plane if the number of vertices of its unit sphere is $ 4n, $ for some $ n \in \mathbb{N}. $ We next obtain a complete characterization of polygonal Radon planes in terms of a tractable geometric concept introduced in this article. It follows from our characterization that every regular polygon with $ 4n+2 $ vertices, where $ n \in \mathbb{N}, $ is the unit sphere of a Radon plane. We further give example of a family of Radon planes for which the unit spheres are hexagons, but not regular ones.