A Geometric Characterization of Polygonal Radon Planes

We study unit circles of polygonal Radon planes from a geometric point of view. In particular, we prove that a two-dimensional real polygonal Banach space X cannot be a Radon plane if the number of vertices of its unit circle is 4n, for some natural number n. Also we 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 is a natural number, is the unit circle of a Radon plane. Furthermore, we describe types of Radon planes for which the unit circles are hexagons, but not regular ones.