On the solvability of the discrete analogue of the Minkowski–Alexandrov problem

The article deals with the multidimensional discrete analogue of the Minkowski problem in the production of A. D. Aleksandrov on the existence of a convex polyhedron with given curvatures at the vertices. We find the conditions for the solvability of this problem in a general setting, when the curvature measure at the polyhedron vertices is defined by an arbitrary continuous function defined on a field $F: \mathbb S^{n-1}\to (0,+\infty)$. The basis for solving the problem is the solvability of the problem whether each triangulation of a finite set of points $ P \subset \mathbb S^{n-1} $ of the unit sphere corresponds a convex polyhedron whose faces normal belong to the set $ P $.