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

TL;DR

This paper characterizes polygonal Radon planes by their unit sphere shapes, showing that regular polygons with 4n+2 vertices form Radon planes and providing examples with hexagon unit spheres that are not regular.

Contribution

It provides a complete geometric characterization of polygonal Radon planes, identifying which regular polygons can serve as their unit spheres.

Findings

Polygonal Radon planes cannot have unit spheres with 4n vertices.

Regular polygons with 4n+2 vertices are unit spheres of Radon planes.

Examples of Radon planes with hexagon unit spheres that are not regular.

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.