page_title: Hilbert's fourth problem
section_title: Hilbert's geometry
hierarchical_section_title: Hilbert's fourth problem / Prehistory of  Hilbert's fourth problem / Hilbert's geometry
context_page_description: In mathematics, Hilbert's fourth problem in the 1900 Hilbert 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, 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 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 Hilbert. Nevertheless, a solution was sought with the German mathematician Georg Hamel being the first who 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.
context_section_description: Let  be a bounded open convex set with  the boundary of  class C² and positive normal curvatures. Similarly to the Lobachevsky space, the hypersurface  is called the absolute of Hilbert's geometry.
  Hilbert's distance (see fig.) is  defined by
  The distance  induces the Hilbert–Finsler metric  on U. For any   and  (see fig.), we have
  The metric is symmetric and flat. In 1895, Hilbert introduced this metric as a generalization of  the  Lobachevsky geometry. If the hypersurface  is an ellipsoid, then we have the Lobachevsky geometry.
caption: Hilbert–Finsler metric