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Some remarks on the Finslerian version of Hilbert’s fourth problem. (English) Zbl 1228.53085
The article discusses the Finslerian formulation of Hilbert’s fourth problem: the problem of finding Finsler functions on \(T^0\mathbb R^n\) whose geodesics, as point sets, are straight lines. The author considers both cases: positively and absolutely homogeneous functions. One of the goals of the present paper is to make two different ways of characterizing projective Finsler spaces compatible. The classical approach is Hamel’s and Rapcsak’s versions of Hilbert’s fourth problem. A new approach is due to Alvarez Paiva. He proved that projective absolutely homogeneous Finsler functions correspond to symplectic structures on the space of orientated lines in \(\mathbb R^n\) with certain properties. In this paper, the author proves Paiva’s results using more elementary methods and establishes the relation between Paiva’s work and Hamel’s approach, in the case of absolutely homogeneity. The article contains also a detailed discussion of strong convexity.

53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)
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