In this paper we consider two notions that have been discovered and
rediscovered by geometers and analysts since 1917 up to the present day: Radon
curves and antinorms.
A Radon curve is a special kind of centrally symmetric closed convex curve in
the plane. A Radon plane is a normed plane obtained by using a Radon curve as
(the boundary of) the unit ball. Many known results in Euclidean geometry also
hold for Radon planes, for example the triangle and parallelogram area
formulas, certain theorems on angular bisectors, the area formula of a polygon
circumscribed about a circle, certain isoperimetric inequalities, and the
non-expansive property of certain non-linear projections.
These results may be further generalized to arbitrary normed planes if we
formally change the statement of the result by referring in some places to the
antinorm instead of the norm. The antinorm is a norm dual to the norm of an
arbitrarily given normed plane, although it lives in the same plane as the
original norm.
It is the purpose of this mainly expository paper to give a list of results
on antinorms that generalize results true for Radon norms, and in many cases
characterize Radon norms among all norms in the plane. Many of the results are
old, well-known, and have often been rediscovered. However, for most of the
results we give streamlined proofs. Also, some of the characterizations of
Radon curves given here seem not to have appeared previously in print.Comment: 24 pages, 7 figure