A Remarkable Quadratic Form of Voronoi (original) (raw)

Abstract

At the end of his posthumous memoir, Voronoi defined a positive quadratic form, which he denoted by \(\omega(x)\). Voronoi proved that this form lies on an extreme ray of a simplicial \(L\)-domain adjacent along a facet to the principal \(L\)-domain. In this paper, it is shown that the form \(\omega\) naturally arises as a metric form of the lattices \(L^{n+1}_{Z,D}(h_{n-1})\), where \(h_{n-1}^2=(n-2)/4\). These lattices are superpositions of layers of the cubic lattice \(Z^n\) and the root lattice \(D_n\). In the case of \(D_n\) of odd dimension \(n\), the lattice \(L^{n+1}_D(h_{n-1})\) has an extremal Delaunay polytope. For \(n=5\), the lattice \(L^6_D(h_4)\), where \(h_4^2=3/4\), is isomorphic to the root lattice \(E_6\).

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References

  1. N. P. Dolbilin, “Properties of faces of parallelohedra,” Proc. Steklov Inst. Math. 266, 105–119 (2009).
    Article MathSciNet Google Scholar
  2. S. S. Ryshkov and E. P. Baranovskii, “\(C\)-types of \(n\)-dimensional lattices and 5-dimensional primitive parallelohedra (with application to the theory of coverings),” Proc. Steklov Inst. Math. 137, 1–140 (1976).
    Google Scholar
  3. S. S. Ryshkov, “The structure of primitive parallelohedra and Voronoi’s last problem,” Russ. Math. Surveys 53 (2), 403–405 (1998).
    Article Google Scholar
  4. V. P. Grishukhin, “Contact vectors of point lattices,” Math. Notes 113 (5), 642–649 (2023).
    Article MathSciNet Google Scholar
  5. G. F. Voronoi, “A study of primitive parallelohedra,” in Collected Works in Three Volumes, Vol. 2 (Izd. Akademii Nauk Ukrainskoi SSR, Kiev, 1952), pp. 239–368 [in Russian].
    Google Scholar
  6. M. Dutour, “Infinite series of extreme Delaunay polytopes,” European J. Combin. 26 (1), 129–132 (2005).
    Article MathSciNet Google Scholar
  7. J. Conway and N. Sloane, Sphere Packings, Lattices, and Groups (Springer-Verlag, New York, 1999).
    Book Google Scholar
  8. E. P. Baranovskii, “Partitioning of Euclidean spaces into \(L\)-polytopes of some perfect lattices,” Proc. Steklov Inst. Math. 196, 29–51 (1992).
    Google Scholar
  9. J. H. Conway and N. J. A. Sloane, “The cell structures of certain lattices,” in Miscellanea Mathematica (Springer-Verlag, Berlin, 1991), pp. 71–107.
    Chapter Google Scholar
  10. S. S. Ryshkov, “A direct geometric description of the \(n\)-dimensional Voronoi parallelohedra of second type,” Russian Math. Surveys 54 (1), 264–265 (1999).
    Article MathSciNet Google Scholar
  11. V. P. Grishukhin, “The Voronoi polyhedra of the rooted lattice \(E_6\) and of its dual lattice,” Discrete Math. Appl. 21 (1), 91–108 (2011).
    Article MathSciNet Google Scholar

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This work was supported by ongoing institutional funding. No additional grants to carry out or direct this particular research were obtained.

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  1. Central Economics and Mathematics Institute of Russian Academy of Sciences, Moscow, 117418, Russia
    V. P. Grishukhin

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Correspondence toV. P. Grishukhin.

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Translated from Matematicheskie Zametki, 2025, Vol. 117, No. 1, pp. 48–61 https://doi.org/10.4213/mzm14300.

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Grishukhin, V.P. A Remarkable Quadratic Form of Voronoi.Math Notes 117, 51–61 (2025). https://doi.org/10.1134/S0001434625010055

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