For citation:
Klyachin V. A. On the Solvability of the Discrete Analogue of the Minkowski – Alexandrov Problem. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2016, vol. 16, iss. 3, pp. 281-288. DOI: 10.18500/1816-9791-2016-16-3-281-288, EDN: WMIIGN
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 : S n−1 → (0, +∞). The basis for solving the problem is the solvability of the problem whether each triangulation of a finite set of points P ⊂ S n−1 of the unit sphere corresponds a convex polyhedron whose faces normal belong to the set P.
- Pogorelov A. V. Mnogomernaia problema Minkovskogo [Multidimensional Minkowsky problem]. Moscow, Nauka, 1971, 95 p. (in Russian).
- Iorgens K. Uber die Losungen der Differentialgleichung nt − s 2 = 1. Math. Ann., 1954, vol. 127, pp. 130–134.
- Calabi E. Improper affine hypersheres of convex type and a generalizations of theorem by K. Iorgens. ¨ Michigan Math. J., 1958, vol. 5, iss. 2, pp. 105–126. DOI: https://doi.org/10.1307/mmj/1028998055.
- Aleksandrov A. D. Dirichlet problem for equation Det||zij || = ϕ(z1, ..., zn, z, x1, ..., xn). I. Vestnik LGU, Ser. Mathematics, mechanics and astronomy, 1958, no. 1, iss. 1, pp. 5–24 (in Russian).
- Bodrenko A. I. The solution of the Minkowski problem for open surfaces in Riemannian space. Arxiv.org, 2007, arXiv:0708.3929.
- Aleksandrov A. D. Vypuklye mnogogranniki [Convex polyhedra]. Moscow ; Leningrad, GITTL, 1950, 429 p. (in Russian).
- Zil’berberg A. A. On existence of closed convex polyhedra with prescrobed curvature of vertexes. Uspehi Mat. Nauk, 1962, vol. 17, no. 4(106), pp. 119–126 (in Russian).
- 995 reads