The ratio of the length of the unit circle to the area of the unit disc in Minkowski planes
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. © 2004 American Mathematical Society.
Proceedings of the American Mathematical Society
Mustafaev, Zokhrab, "The ratio of the length of the unit circle to the area of the unit disc in Minkowski planes" (2005). Faculty Articles Indexed in Scopus. 1951.