of flat metrics. On a set of lines on the projective plane he introduced a completely additive non-negative measure , which satisfies the following conditions: , where is a set of straight lines passing through a point P; , where is a set of straight lines passing through some set X that contains a straight line segment; is finite. If we consider a -metric in an arbitrary convex domain of a projective space , then condition 3) should be replaced by the following: for any set H such that H is contained in and the closure of H does not intersect the boundary of , the inequality holds. Using this measure, the -metric on is defined by where is the set of straight lines that intersect the segment . The triangle inequality for this metric follows from Pasch's theorem. Theorem. -metric on is flat, i.e., the geodesics are the straight lines of the projective space. But Busemann was far from the idea that -metrics exhaust all flat metrics. He wrote, "The freedom in the choice of a metric with given geodesics is for non-Riemannian metrics so great that it may be doubted whether there really exists a convincing characterization of all Desarguesian spaces". Two-dimensional case Pogorelov's theorem The following theorem was proved by Pogorelov in 1973 Theorem. Any two-dimensional continuous complete flat metric is a -metric.Thus Hilbert's fourth problem for the two-dimensional case was completely solved. A consequence of this is that you can glue boundary to boundary two copies of the same planar convex shape, with an angle twist between them, you will get a 3D object without crease lines, the two faces being developable. Ambartsumian's proofs In 1976, Ambartsumian proposed another proof of Hilbert's fourth problem. His proof uses the fact that in the two-dimensional case the whole measure can be restored by its values on biangles, and thus be defined on triangles in the same way as the area of a triangle is defined on a sphere. Since the triangle inequality holds, it follows that this measure is positive on non-degenerate triangles and is determined on all Borel sets. However, this structure can not be generalized to higher dimensions because of Hilbert's third problem solved by Max Dehn. In the two-dimensional case, polygons with the same volume are scissors-congruent. As was shown by Dehn this is not true for a higher dimension. Three dimensional case For three