1890, for solving problems on the theory of numbers, Hermann Minkowski introduced a notion of the space that nowadays is called the finite-dimensional Banach space. Minkowski space Let be a compact convex hypersurface in a Euclidean space defined by where the function satisfies the following conditions: and the form is positively definite. The length of the vector OA is defined by: A space with this metric is called Minkowski space. The hypersurface is convex and can be irregular. The defined metric is flat. Finsler spaces Let M and be a smooth finite-dimensional manifold and its tangent bundle, respectively. The function is called Finsler metric if ; For any point the restriction of on is the Minkowski norm. is Finsler space. Hilbert's geometry 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. Funk metric In 1930, Funk introduced a non-symmetric metric. It is defined in a domain bounded by a closed convex hypersurface and is also flat. σ-metrics Sufficient condition for flat metrics Georg Hamel was first to contribute to the solution of Hilbert's fourth problem. He proved the following statement. Theorem. A regular Finsler metric is flat if and only if it satisfies the conditions: Crofton formula Consider a set of all oriented lines on a plane. Each line is defined by the parameters and where is a distance from the origin to the line, and is an angle between the line and the x-axis. Then the set of all oriented lines is homeomorphic to a circular cylinder of radius 1 with the area element . Let be a rectifiable curve on a plane. Then the length of is where is a set of lines that intersect the curve , and is the number of intersections of the line with . Crofton proved this statement in 1870. A similar statement holds for a projective space. Blaschke–Busemann measure In 1966, in his talk at the International Mathematical Congress in Moscow, Herbert Busemann introduced a new class