Hilbert's fourth problem
In mathematics, Hilbert's fourth problem in the 1900 list of Hilbert's problems is a foundational question in geometry. In one statement derived from the original, it was to find — up to an isomorphism — all geometries that have an axiomatic system of the classical geometry (Euclidean, hyperbolic and elliptic), with those axioms of congruence that involve the concept of the angle dropped, and `triangle inequality', regarded as an axiom, added. If one assumes the continuity axiom in addition, then, in the case of the Euclidean plane, we come to the problem posed by Jean Gaston Darboux: "To determine all the calculus of variation problems in the plane whose solutions are all the plane straight lines." There are several interpretations of the original statement of David Hilbert. Nevertheless, a solution was sought, with the German mathematician Georg Hamel being the first to contribute to the solution of Hilbert's fourth problem. A recognized solution was given by Ukrainian mathematician Aleksei Pogorelov in 1973. In 1976, Armenian mathematician Rouben V. Ambartzumian proposed another proof of Hilbert's fourth problem. (Wikipedia).
