Israel Journal of Mathematics

, Volume 103, Issue 1, pp 41–65 | Cite as

Parametric versions of Hilbert’s fourth problem

  • R. V. Ambartzumian
  • V. K. Oganian


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.


Support Function Orientation Function Convex Domain Level Line Riemann Metrics 
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Copyright information

© Hebrew University 1998

Authors and Affiliations

  1. 1.Institute of Mathematics, National Academy of Sciences of ArmeniaYerevan State UniversityYerevanArmenia

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