Geodesic
The neighborhood of the Voronoi main perfect form from five variables
Čebyševskij sbornik, Tome 24 (2023) no. 1, pp. 219-227.

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Voronoi obtained three results for perfect forms. First, he proved that the form corresponding to the closest packing is perfect. Secondly, he established that there are a finite number of perfect forms from a given number of variables. And most importantly, thirdly, Voronoi proposed a method for finding all perfect forms. This method relies on the so-called perfect polyhedron, a highly complex multidimensional polyhedron introduced by Voronoi. In principle, having found all perfect forms by the Voronoi method, one can calculate the densities for a finite number of corresponding packings and single out those that correspond to the maximum value. The classical Voronoi problem of finding perfect forms, closely related to Hermite's well-known problem of arithmetic minima of positive quadratic forms. They also appeared in the works of S.L. Sobolev and Kh.M. Shadimetov in connection with the construction of lattice optimal cubature formulas. In this paper, we propose an improved Voronoi algorithm for calculating the Voronoi neighborhood of a perfect form in many variables, and using this algorithm, the Voronoi neighborhood of the main perfect form in five variables is calculated.
Keywords: densest packing, perfect forms, Voronoi algorithm, multidimensional polyhedron. 
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O. Kh. Gulomov. The neighborhood of the Voronoi main perfect form from five variables. Čebyševskij sbornik, Tome 24 (2023) no. 1, pp. 219-227. http://geodesic.mathdoc.fr/item/CHEB_2023_24_1_a15/

[1] G. F. Voronoi, “Some properties of positive quadratic forms”, Own. cit., v. 2, Publishing house of the Academy of Sciences of the Ukrainian SSR, Kiev, 1952, 171–238

[2] E. S. Barnes, “The complete enumeration of perfect snare forms”, Phil. Trans. Rog. Soc. London, A-249 (1957), 461–506 | MR | Zbl

[3] O. Kh. Gulomov, S. Yu. Shodiev, “Calculation of perfect forms in four variables using the advanced Voronoi algorithm”, Chebyshevskii sbornik, 13:2 (2012), 59–63 | MR | Zbl

[4] S. S. Ryshkov, Basic extremal problems of the geometry of positive quadratic forms, Doctoral dissertation, M., 1970, 171 pp. | MR

[5] M. M. Anzin, “The density of a lattice covering for n = 11 and n = 14”, Uspekhi Mat. Nauk, 57:2 (2002), 187–188 | DOI | MR | Zbl

[6] O. Kh. Gulomov, “Algorithms for constructing a perfect gonohedron based on the duality principle from the theory of linear inequalities”, Uzbek mathematical journal, 2001, no. 2, 31–36 | MR

[7] O. Kh. Gulomov, S. Yu. Shodiev, “Calculation of perfect forms from four variables using the improved Voronoi algorithm”, Chebyshevskii sbornik, 13:2 (2014), 59–63 | MR

[8] O. Kh. Gulomov, S. Yu. Shodiev, “About necessary and sufficient condition for strong stationarity of the positive quadratic form”, Math. Forum, 9:6 (2014), 267–272 | MR

[9] O. Kh. Gulomov, S. Yu. Shodiev, “On an Algorithm for Finding Integer Points on Perfect Ellipsoids”, AIP Conference Proceedings, 2365, 2021, 050001, 6 pp. | DOI

[10] O. Kh. Gulomov, B. A.Khudayarov, K. Sh. Ruzmetov, F. Zh. Turaev, “Quadratic forms related to the voronoi$\mathrm{\Leftrightarrow }$s domain faces of the second perfect form in seven variables”, Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications and Algorithms, 28 (2021), 15–23 | MR | Zbl

[11] J. Martinet, Perfect lattices in Euclidean spaces, Springer, 2003 | MR | Zbl

[12] C. Soule, “Perfect forms and the Vandiver conjecture”, J. Reine Angew. Math., 517 (1999), 209–221 | DOI | MR | Zbl

[13] Dutour Sikiric M., Vallentin F., Schurmann A., “Classification of eight-dimensional perfect forms”, Electronic Research Inducement's of the AMS, 13 (2007), 21–32 | MR | Zbl

[14] Dutour Sikiric M., Sch?urmann A., Vallentin F., “Complexity and algorithms for computing Voronoi cells of lattices”, Math. Comp., 78 (2009), 1713–1731 | DOI | MR | Zbl

[15] Sobolev S.L., Introduction to the theory of cubature formulas, Nauka, M., 1974, 808 pp. | MR