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
kalidas.mandal14@gmail.com
Debmalya Sain
Indian Institute of Science, Bengaluru 560012, Karnataka, India
saindebmalya@gmail.com
Kallol Paul
Department of Mathematics, Jadavpur University, Kolkata 700032, India
kalloldada@gmail.com
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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