Angular measures and Birkhoff orthogonality in Minkowski planes

Let x and y be two unit vectors in a normed plane R 2. 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 (Fankhänel 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.