Hilbert's Fourth Problem (on the straight line as the shortest distance between two points) (Q2736240)

Problem IV was to find a geometry which is different by ``close'' to (in a specified sense) Euclidean geometry. In 1901-1903 G. Hamel has shown that typical examples of such a geometry were already found by H. Minkowski and D. Hilbert himself. The answer has satisfied Hilbert but in 1929 L. Berwald and P. Funk proposed a different understanding of the problem, thus initiating a new line of research. The author reports the latter story up to the book of \textit{A. V. Pogorelov} [A complete solution of Hilbert's Fourth Problem, Sov. Math., Dokl. 14, 46-49 (1973); translation from Dokl. Akad. Nauk SSSR 208, 48-51 (1973; Zbl 0285.52007)] and its review by \textit{H. Buseman} [Problem IV: Desarguesian spaces, in: Math. Dev. Hilbert Probl., Proc. Symp. Pure Math. 28, 131-141 (1976; Zbl 0352.50010)].NEWLINENEWLINEFor the entire collection see [Zbl 0902.00028].