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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
Article

Abstract

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.

Keywords

Support Function Orientation Function Convex Domain Level Line Riemann Metrics 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    R. Alexander,Planes for which the lines are the shortest paths between points, Illinois Journal of Mathematics22 (1978), 177–190.zbMATHMathSciNetGoogle Scholar
  2. [2]
    R. V. Ambartzumian,A note on pseudo-metrics on the plane, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete29 (1974), 25–31.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    R. V. Ambartzumian,Combinatorial Integral Geometry with Applications to Mathematical Stereology, Wiley, Chichester, 1982.zbMATHGoogle Scholar
  4. [4]
    R. V. Ambartzumian,Factorization Calculus and Geometric Probability, Cambridge University Press, Cambridge, 1990.zbMATHGoogle Scholar
  5. [5]
    R. V. Ambartzumian with the Appendix by V. K. Oganian,Measure generation by Euler functionals, Advances in Applied and Probability (SGSA)27 (1995), 606–626.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    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
  7. [7]
    A. V. Pogorelov,Hilbert’s Fourth Problem, Winston & Sons, London, 1979.Google Scholar

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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