A new construction of Radon curves and related topics

TL;DR

This paper introduces a novel convexity-based method for constructing Radon curves, avoiding traditional Euclidean and Minkowski geometry techniques, and explores properties and characterizations of Radon curves in normed planes.

Contribution

It provides a new convexity-centric construction of Radon curves and characterizations solely using convex geometry, diverging from classical and Minkowski approaches.

Findings

Radon curves can be constructed using convexity methods only.

Properties of normed planes with Radon unit circles are analyzed.

Characterizations of Radon curves are given in convex geometric terms.

Abstract

We present a new construction of Radon curves which only uses convexity methods. In other words, it does not rely on an auxiliary Euclidean background metric (as in the classical works of J. Radon, W. Blaschke, G. Birkhoff, and M. M. Day), and also it does not use typical methods from plane Minkowski Geometry (as proposed by H. Martini and K. J. Swanepoel). We also discuss some properties of normed planes whose unit circle is a Radon curve and give characterizations of Radon curves only in terms of Convex Geometry.