反范数与Radon曲线
Antinorms and Radon curves
摘要：
在本文中，我们考虑两个概念，已经发现和重新发现的几何学家和分析家自1917年至今：氡曲线和反范数。Radon曲线是平面上一类特殊的中心对称闭凸曲线。Radon平面是通过使用Radon曲线作为单位球（的边界）而获得的赋范平面。欧几里得几何中的许多已知结果也适用于拉东平面，例如三角形和三角形面积公式，角平分线的某些定理，外切圆的多边形的面积公式，某些等周不等式，以及某些非线性投影的非扩张性。这些结果可以进一步推广到任意赋范平面，如果我们正式改变的结果，在某些地方提到的antinertium，而不是规范的声明。逆模是一个与任意给定的赋范平面的范数对偶的范数，尽管它与原范数生活在同一平面上。这主要是一个简要的文件的目的是给一个列表的反范数的结果，推广结果真实的氡规范，并在许多情况下，特点氡规范之间的所有规范在平面上。许多结果是旧的，众所周知的，并经常被重新发现。然而，对于大多数结果，我们给出了简化的证明。此外，这里给出的一些Radon曲线的特征似乎以前没有出现在印刷品中。
英文摘要：
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.
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