dimensional case Pogorelov proved the following theorem. Theorem. Any three-dimensional regular complete flat metric is a -metric.However, in the three-dimensional case -measures can take either positive or negative values. The necessary and sufficient conditions for the regular metric defined by the function of the set to be flat are the following three conditions: the value on any plane equals zero, the value in any cone is non-negative, the value is positive if the cone contains interior points. Moreover, Pogorelov showed that any complete continuous flat metric in the three-dimensional case is the limit of regular -metrics with the uniform convergence on any compact sub-domain of the metric's domain. He called them generalized -metrics. Thus Pogorelov could prove the following statement. Theorem. In the three-dimensional case any complete continuous flat metric is a -metric in generalized meaning.Busemann, in his review to Pogorelov’s book "Hilbert’s Fourth Problem" wrote, "In the spirit of the time Hilbert restricted himself to n = 2, 3 and so does Pogorelov. However, this has doubtless pedagogical reasons, because he addresses a wide class of readers. The real difference is between n = 2 and n>2. Pogorelov's method works for n>3, but requires greater technicalities". Multidimensional case The multi-dimensional case of the Fourth Hilbert problem was studied by Szabo. In 1986, he proved, as he wrote, the generalized Pogorelov theorem. Theorem. Each n-dimensional Desarguesian space of the class , is generated by the Blaschke–Busemann construction. A -measure that generates a flat measure has the following properties: the -measure of hyperplanes passing through a fixed point is equal to zero; the -measure of the set of hyperplanes intersecting two segments [x, y], [y, z], where x, y та z are not collinear, is positive. There was given the example of a flat metric not generated by the Blaschke–Busemann construction. Szabo described all continuous flat metrics in terms of generalized functions. Hilbert's fourth problem and convex bodies Hilbert's fourth problem is also closely related to the properties of convex bodies. A convex polyhedron is called a zonotope if it is the Minkowski sum of segments. A convex body which is a limit of zonotopes in the Blaschke – Hausdorff metric is called zonoid. For zonoids, the support function is represented by where is an even positive Borel measure on a sphere . The Minkowski space is generated by the Blaschke–Busemann construction if and only if the support function of the