Total Angle around a Point in Minkowski Plane

Abstract

A new angular measure in Minkowski plane was introduced recently. It is additive, invariant under invertible linear transformations, and has a clear interpretation as an "amount of rotation" from one direction to another one. The total angle τ around a point depends on the unit ball. It is known that \(4.443 \approx \sqrt{2}\pi \leq \tau \leq 8\). We show that \(4.985 \approx 4 \sqrt{2}\, {\rm ln}\,{\rm tan} \frac{3\pi}{8} \leq \tau \leq 8\, {\rm ln}\,{\rm tan} \frac{3\pi}{8}\, \approx 7.051\).