While the general impression is that Finsler geometry is a complicated and computation-intensive field, the reality is that, like convex, metric, and integral geometry, many of its examples and constructions are natural, elementary, and non-trivial. No doubt this assertion will come as a surprise to some and therefore I will back it up with a link to a paper on Hilbert's fourth problem that I wrote for advanced undergraduates and beginning graduate students. The paper is based on (part of) a talk I gave at the MASS Colloquium --- a wonderful program where talented undergraduate students from all over the United States converge to Penn State for a semester of advanced mathematics --- and will appear in its proceedings. I have not signed away my copyright yet, so if you see you like the paper print it out soon. If you have time, send me some feedback: it may motivate me to write part II.
I invite other researchers in Finsler and Minkowski geometry to write this sort of expository papers in order to motivate graduate and undergraduate students to learn about Finsler and Minkowski spaces. I would be happy to post their preprints here.
For the Finsler geometer, I mention that hidden in exercise 6.3 is the following (new?) result: if the geodesics of a Finsler metric on the 2-torus coincide as point sets with the geodesics of a flat torus, then the metric is flat.
14th of October, 2002.
Hilbert's fourth problem in two dimensions I
Abstract. Hilbert's fourth problems asks to construct and study the geometries in which the straight line segment is the shortest connection between two points. In this paper the reader shall find an elementary introduction to the problem and its solutions in dimension two by Busemann, Pogorelov, and Ambartzumian. The relationship between integral geometry and inverse problems in variational calculus is emphasized.
On the perimeter and area of the unit disc
Abstract. This is a survey of the most basic results on the geometry of unit discs in two-dimensional normed spaces.
Antinorms and Radon curves
This is a survey on Radon curves and the relationship between circles and solutions of the isoperimetric problem in two-dimensional normed spaces.