A new construction of Radon curves and related topics

V. Balestro, H. Martini and R. Teixeira, A new construction of Radon curves and related topics, Aequationes Mathematicae 90(5), pp. 1013-1024, 2016. (arXiv link) (journal link)

Comments: Radon curves are plane curves which are self-dual. Their importance in Minkowski geometry comes from the fact that they are unit circles of plane norms whose Birkhoff orthogonality relations are symmetric. The classic construction of Radon curves involves an auxiliary Euclidean structure fixed in the plane, and later Martini and Swanepoel found a construction that does not rely in such a structure. Alternatively, they start with a convex curve defined for a quadrant, and define an associated norm. In this paper, we provide a construction of Radon curves based only on basic convexity and vector spaces concepts. A purely geometric characterization of such curves is obtained inspired both by this new construction and by Düvelmeyer’s work on bisectors in Minkowski planes.