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

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 $n\in \mathbb{N}\mathrm{.}$ 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.