Geodesic
On the solvability of the discrete analogue of the Minkowski--Alexandrov problem
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 16 (2016) no. 3, pp. 281-288.

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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 $. 
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V. A. Klyachin. On the solvability of the discrete analogue of the Minkowski--Alexandrov problem. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 16 (2016) no. 3, pp. 281-288. http://geodesic.mathdoc.fr/item/ISU_2016_16_3_a4/

[1] Pogorelov A. V., Multidimensional Minkowsky problem, Nauka, M., 1971, 95 pp. (in Russian)

[2] Iörgens K., “Über die Lösungen der Differentialgleichung $nt-s^2=1$”, Math. Ann., 127 (1954), 130–134 | DOI | MR

[3] Calabi E., “Improper affine hypersheres of convex type and a generalizations of theorem by K. Iörgens”, Michigan Math. J., 5:2 (1958), 105–126 | DOI | MR | Zbl

[4] Aleksandrov A. D., “Dirichlet problem for equation $Det||z_{ij}||=\varphi(z_1,...,z_n,z,x_1,...,x_n)$. I”, Vestnik LGU, Ser. Mathematics, mechanics and astronomy, 1958, no. 1(1), 5–24 (in Russian)

[5] Bodrenko A. I., The solution of the Minkowski problem for open surfaces in Riemannian space, 2007, arXiv: 0708.3929 [math.DG]

[6] Aleksandrov A. D., Convex polyhedra, GITTL, M.–L., 1950, 429 pp. (in Russian)

[7] Zil'berberg A. A., “On existence of closed convex polyhedra with prescrobed curvature of vertexes”, Uspehi Mat. Nauk, 17:4(106) (1962), 119–126 (in Russian) | MR | Zbl