The ratio of the length of the unit circle to the area of the unit disc in Minkowski planes

Abstract

In their paper “An Introduction to Finsler Geometry,” J. C. Alvarez and C. Duran asked if there are other Minkowski planes besides the Euclidean for which the ratio of the Minkowski length of the unit “circle” to the Holmes-Thompson area of the unit disc equals 2. In this paper we show that this ratio is greater than 2, and that the ratio 2 is achieved only for Minkowski planes that are affine equivalent to the Euclidean plane. In other words, the ratio is 2 only when the unit “circle” is an ellipse.