Angular measures and Birkhoff orthogonality in Minkowski planes
- PDF / 314,323 Bytes
- 9 Pages / 439.37 x 666.142 pts Page_size
- 54 Downloads / 183 Views
Aequationes Mathematicae
Angular measures and Birkhoff orthogonality in Minkowski planes ´rton Naszo ´ di , Vilmos Prokaj, and Konrad Swanepoel Ma Abstract. Let x and y be two unit vectors in a normed plane R2 . We say that x is Birkhoff orthogonal to y if the line through x in the direction y supports the unit disc. A B-measure (Fankh¨ anel in Beitr Algebra Geom 52(2):335–342, 2011) is an angular measure μ on the unit circle for which μ(C) = π/2 whenever C is a shorter arc of the unit circle connecting two Birkhoff orthogonal points. We present a characterization of the normed planes that admit a B-measure. Mathematics Subject Classification. Primary 52A21; Secondary 28A75, 46B20. Keywords. Angle measure, Birkhoff orthogonality, Minkowski space, Normed space, Radon planes.
1. Introduction Let K be an origin-symmetric convex body in the plane, that is, a compact convex set with non-empty interior in R2 , and consider the normed plane (R2 , ·K ), where xK = min {λ > 0 : x ∈ λK} for any x ∈ R2 . Then K is the unit ball of the norm, and its boundary bd K the unit circle. Let x, y ∈ bd K be two unit vectors in R2 . We say that x is Birkhoff orthogonal to y, and denote it by x y, if xK ≤ x + tyK for all t ∈ R. Geometrically, this means that the line through the point x in the direction y supports the unit ball K. In general, Birkhoff orthogonality is not a symmetric relation. Normed planes where Birkhoff orthogonality is symmetric are called Part of the research was carried out while MN was a member of J´ anos Pach’s chair of DCG at EPFL, Lausanne, which was supported by Swiss National Science Foundation Grants 200020-162884 and 200021-175977, and while MN and KS visited the Mathematical Research Institute Oberwolfach in their Research in Pairs programme. MN also acknowledges the support of the National Research, Development and Innovation Fund Grant K119670.
´ di et al. M. Naszo
AEM
Radon planes and the boundaries of their unit balls Radon curves (see the survey [5]). A Borel measure μ on bd K is called an angular measure, if μ(bd K) = 2π, μ(X) = μ(−X) for every Borel subset X of bd K, and μ is continuous, that is, μ({x}) = 0 for every x ∈ bd K. There always exists an angular measure on bd K, such as the one-dimensional Hausdorff measure on bd K normalized to 2π, but an arbitrary angular measure does not necessarily have any relation to the geometry of (R2 , ·K ). A natural problem then is to find angular measures with interesting geometric properties. For instance, Brass [2] showed that whenever the unit ball is not a parallelogram, there is an angular measure in which the angles of any equilateral triangle are equal. This type of angular measure is very useful in studying packings of unit balls [2,8]. Angular measures with other properties have been proposed; see the survey [1, Section 4] for an overview. An angular measure μ is called a B-measure [3] if μ(C) = π/2 for every closed arc C of bd K that contains no opposite points of bd K, and whose endpoints x and y satisfy x y. The main result of this note (
Data Loading...
