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

V. A. Klyachin

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 16 (2016) no. 3, pp. 281-288

Résumé

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 $.