by an outstanding geometer A.D. Aleksandrov. He developed the theory of metric spaces of non-negative curvature, so-called Aleksandrov metric spaces. As a special case, the theory covered the intrinsic geometry of general convex surfaces, that is boundaries of convex bodies. Aleksandrov studied connections between the intrinsic and extrinsic geometries of general convex surfaces. He proved that every metric of non-negative curvature given on a two-dimensional sphere (including non-smooth metrics, so-called inner metrics) can be isometrically immersed into the three-dimensional Euclidean space in a form of a closed convex surface, but the answers to the following fundamental questions were unknown: is this immersion unique up to rigid motion? if the metric given on the sphere is a regular one and of positive Gaussian curvature, is it true then that the surface with this metric is regular? G. Minkowski proved an existence theorem for a closed convex surface with the Gaussian curvature given as a function of a unit normal under some natural condition on this function; the open question was: if the function is regular on a sphere, is the surface regular itself? After solving these problems, the theory created by Aleksandrov would have received “full citizenship” in mathematics and could be applied also in the classical regular case. Each of these 3 questions was answered positively by A.V. Pogorelov. Using synthetic geometric methods, he developed geometric methods to obtain priori estimates for solutions of Monge-Ampère equations. On the one hand, he used these equations to solve geometric problems; on the other hand, based on geometric reasons, he constructed a generalized solution of a Monge-Ampère equation and then proved its regularity for a regular right-hand side of the equation. In fact, in these pioneering works A.V. Pogorelov laid the foundation of the field of geometric analysis. He proved the following fundamental results: Let F1 and F2 be two closed convex isometric surfaces in the three-dimensional Euclidean space or in a spherical space. Then the surfaces coincide up to the rigid motion. A closed convex surface in a space of constant curvature is rigid outside flat domains on it. This means that the surface admits only trivial infinitesimal bendings. If the metric of a convex surface is regular of regularity Ск, k≥2, in a space of constant curvature К* and the Gaussian curvature of the surface satisfies К>К*, then the surface is Ск-1,α. For domains on convex surfaces assertions 1) and 2)