Antinorms and Radon curves

@article{Martini2006AntinormsAR,
  title={Antinorms and Radon curves},
  author={Horst Martini and Konrad J. Swanepoel},
  journal={aequationes mathematicae},
  year={2006},
  volume={72},
  pages={110-138}
}
Summary.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… 
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References

SHOWING 1-10 OF 66 REFERENCES
Stability of Blaschke's Characterization of Ellipsoids and Radon Norms
  • P. Gruber
  • Mathematics
    Discret. Comput. Geom.
  • 1997
TLDR
This article proves quantitative versions of the following statements: If each shadow boundary of a convex body in Ed under parallel illumination contains a curve which is almost planar and has the same shadow as the shadow boundary, then the body is approximately ellipsoidal.
On minimum circumscribed polygons
This paper contains the proofs of two theorems on the n-gon Mn of minimum area circumscribed about a convex region R in the plane. Theorem 1 shows that the area of Mn is a convex function of n and
Equiframed Curves – A Generalization of Radon Curves
Abstract.Equiframed curves are centrally symmetric convex closed planar curves that are touched at each of their points by some circumscribed parallelogram of smallest area. These curves and their
Orthogonality and linear functionals in normed linear spaces
The natural definition of orthogonality of elements of an abstract Euclidean space is that x ly if and only if the inner product (x, y) is zero. Two definitions have been given [11](2) which are
The Fermat–Torricelli Problem in Normed Planes and Spaces
We investigate the Fermat–Torricelli problem in d-dimensional real normed spaces or Minkowski spaces, mainly for d=2. Our approach is to study the Fermat–Torricelli locus in a geometric way. We
On the Geometry of Simplices in Minkowski Spaces
Let T be a d-dimensional simplex in a d-dimensional real normed space (= Minkowski space). We introduce a special Minkowskian area-measure and Minkowskian trilinear coordinates with respect to T,
Angle Measures and Bisectors in Minkowski Planes
Abstract We prove that a Minkowski plane is Euclidean if and only if Busemann's or Glogovskij's definitions of angular bisectors coincide with a bisector defined by an angular measure in the sense of
Orthogonality and Proportional Norms
Two norms on a real vectorspace define the same orthogonality relation iff they are proportional. The aim of this note is to give a proof of this statement with a minimum of results on convex sets,
Erds Distance Problems in Normed Spaces
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