Resumen de Antinorms and Radon curves

Horst Martini, Konrad J. Swanepoel

• A Radon curve can be used as the unit circle of a norm, with the corresponding normed plane called a Radon plane. An antinorm is a special case of the Minkowski content of a measurable set in a Minkowski space. There is a long list of known results in Euclidean geometry that also hold for Radon planes. These results may sometimes be further generalized to arbitrary normed planes if we formally change such a statement by referring in some places to the antinorm instead of the norm. We present a list of such results for antinorms. Although most of these results are well known, we give streamlined proofs, and show which of these results lead to characterizations of Radon norms.

  As new results we prove two characterizations of Radon curves, one in terms of bisectors and the other in terms of triangles circumscribed about circles. We also solve the Zenodorus problem for Minkowski planes, i.e., we characterize the polygons with n sides that have the largest area for a fixed perimeter in any given Minkowski plane.