Parametric versions of Hilbert’s fourth problem
- 62 Downloads
LetH be the class of sufficiently smooth metrics defined on the Euclidean plane for which the geodesics are the usual Euclidean liens. The general problem is to describe all metrics fromH which at each point possess the length indicatrix from a prescribed parametric class of convex figures. As a tool, a differential equation is proposed derived from the “triangular deficit principle” formulated in an earlier paper of R. V. Ambartzumian. The paper demonstrates that for the case where the length indicatrix is segmental this equation leads to a complete solution. Also, there is a partial result stating that in the case of Riemann metrics the orientation of the ellipsi should necessarily be a harmonic function.
KeywordsSupport Function Orientation Function Convex Domain Level Line Riemann Metrics
Unable to display preview. Download preview PDF.
- V. K. Oganian and A. Abdallah,On generation of measures in the space of lines by Finsler metrics, [in Russian], Izvestiya Akademii Nauk Armenii, Matematika [English translation: Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)]27 (1992), 69–80.Google Scholar
- A. V. Pogorelov,Hilbert’s Fourth Problem, Winston & Sons, London, 1979.Google Scholar