Angular measures and Birkhoff orthogonality in Minkowski planes

TL;DR

This paper characterizes the normed planes that admit a B-measure, an angular measure related to Birkhoff orthogonality, providing insights into geometric structures in Minkowski planes.

Contribution

It offers a characterization of normed planes that support a B-measure, linking angular measures to Birkhoff orthogonality in Minkowski geometry.

Findings

Identifies conditions for the existence of B-measures in Minkowski planes

Connects Birkhoff orthogonality with angular measures in normed spaces

Provides a geometric characterization of planes admitting B-measures

Abstract

Let $x$ and $y$ be two unit vectors in a normed plane $\mathbb{R}^2$. 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 (Fankh\"anel 2011) is an angular measure $\mu$ on the unit circle for which $\mu(C)=\pi/2$ 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.