The norm of a Minkowski plane may be defined by a centrally symmetric convex disc \(B\) which is the unit ball with respect to this norm. The boundary \( \partial B\) of \(B\) is called a Radon curve if conjugate diameters of \(B\) make sense. This may be made more precise by the following definition. Let \(x\) and \(y\) be non-zero vectors in the plane. \(x\) is called normal to \(y\) if \(\| x\| \leq \| x+\lambda y\| \) for all \( \lambda \in \mathbb{R} \), where \(\| .\| \) denotes the norm induced by \(B\). Now \(\partial B\) is a Radon curve if and only if this normality is symmetric. In this case the Minkowski plane is also called a Radon plane. To the norm of a Minkowski plane there corresponds a kind of dual norm which is called the antinorm. The unit disc \(I\) of the antinorm may be obtained by taking the polar body of \(B\) with respect to the Euclidean unit circle and rotating it by \(90^\circ \). It turns out that \(\partial B\) is a Radon curve if and only if \(B\) and \(I\) are homothetic.
The paper discusses a series of results in Euclidean geometry which also hold in Radon planes and may be generalized to arbitrary Minkowski planes if in certain places the norm is replaced by the antinorm. This leads to several characterizations of Radon curves. For instance, the radial projection onto \(B\) is non-expansive (with respect to the norm defined by \(B\) ) if and only if \(\partial B\) is a Radon curve. There is also a discussion of the isoperimetric problem for convex \(n\)-gons, the so-called Zenodorus problem. Most of the results of this paper are not new, but are presented from an interesting point of view and with streamlined proofs. However, some of the characterizations of Radon curves and the solution of the Zenodorus problem are new.