Antinorms and Radon curves

In this paper we consider two notions that have been discovered and rediscovered by geometers and analysts since 1917 up to the present day. The first is that of a Radon curve, introduced by Radon in 1917 [50]. It is a special kind of centrally symmetric closed convex curve in the plane. Any centrally symmetric closed convex curve in the plane defines a norm turning the plane into a two-dimensional normed space or Minkowski plane. Finitedimensional normed spaces or Minkowski spaces were introduced by Minkowski in [44], and a special case (the Lp norm for p = 4) was even alluded to in Riemann’s famous Habilitationsvortrag [51]. For a general introduction to Minkowski spaces, see Thompson’s book [57] and the surveys [41, 40]. The norm thus defined by a Radon curve is called a Radon norm, and the corresponding Minkowski plane a Radon plane. Radon planes have many remarkable, almost-Euclidean properties. For a survey on Radon planes, including further results, see [41, Section 6]. The second notion is that of an antinorm. This is a norm dual in a certain sense to the norm of an arbitrary Minkowski plane. It is a special case of the Minkowski content of a set in a Minkowski space introduced by Minkowski [45]. Busemann [8, 9] showed that the circles in the antinorm (anticircles) are the solutions to the isoperimetric problem in a Minkowski plane. He also showed that anticircles are circles (i.e. the antinorm is proportional to the norm) exactly when the circles are Radon curves. The plane with the antinorm turns out to be isometric to the dual normed space of the plane with the original norm. Note however that the antinorm and the norm are defined in the same plane. Since there is no natural way of identifying a vector space and its dual, even in the finite-dimensional case, any identification of the dual normed plane plane with the antinorm must involve some arbitrary choice. In principle one would have to choose an invertible linear transformation, or equivalently, fix a coordinate system (four degrees of freedom in the two-dimensional case). Choosing a Euclidean structure and using polarity is also sufficient (three degrees of freedom). However, we will explain how one only has to choose a unit of area and an orientation (one degree of freedom), since in the plane there exists up to a constant factor only one symplectic bilinear form (a multiple of the determinant) [2].