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Journal of Convex Analysis 26 (2019), No. 4, 1113--1123
Copyright Heldermann Verlag 2019

A Geometric Characterization of Polygonal Radon Planes

Kalidas Mandal
Department of Mathematics, Jadavpur University, Kolkata 700032, India

Debmalya Sain
Indian Institute of Science, Bengaluru 560012, Karnataka, India

Kallol Paul
Department of Mathematics, Jadavpur University, Kolkata 700032, India

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.

Keywords: Radon plane, Birkhoff-James orthogonality, polygonal Banach space.

MSC: 46B20; 52A21

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