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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     title = {The neighborhood of the {Voronoi} main perfect form from five variables},
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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/