A Restoration Surfaces With Vertices Based on Given External Curvature

A surface with vertices is defined as a convex surface whose regularity is broken at certain points. The problem of restoration of a surface based on a given external curvature was posed and solved by A. D. Alexandrov in the class of convex polyhedra. This problem has also been solved for regular surfaces. In this article, the problem is considered for surfaces exhibiting a violation of regularity at specific points within their domain of definition. The problem is solved when the external curvature is given over a convex domain in the plane and within the framework of the solution to the Monge-Ampère equation for the surface under consideration. In this case, the vertices are projected onto the given points in the plane.