staple
With cedram.org

Search the site

Table of contents for this issue | Previous article | Next article
Mathieu Dutour Sikirić; Konstantin Rybnikov
Delaunay polytopes derived from the Leech lattice
Journal de théorie des nombres de Bordeaux, 26 no. 1 (2014), p. 85-101, doi: 10.5802/jtnb.860
Article PDF | Reviews MR 3232768 | Zbl 06304182
Class. Math.: 11H31, 11H55

Résumé - Abstract

A Delaunay polytope in a lattice $L$ is perfect if any affine transformation that preserve its Delaunay property is a composite of an homothety and an isometry. Perfect Delaunay polytopes are rare in low dimension and here we consider the ones that one can get in lattice that are sections of the Leech lattice.

By doing so we are able to find lattices with several orbits of perfect Delaunay polytopes. Also we exhibit Delaunay polytopes which remain Delaunay in some superlattices. We found perfect Delaunay polytopes with small automorphism group relative to the automorphism group of the lattice. And we prove that some perfect Delaunay polytopes have lamination number $5$, which is higher than previously known $3$.

A well known construction of centrally symmetric perfect Delaunay polytopes uses a laminated construction from an antisymmetric perfect Delaunay polytope. We fully classify the types of perfect Delaunay polytopes that can occur.

Finally, we derived an upper bound for the covering radius of $\Lambda _{24}(v)^{*}$, which generalizes the Smith bound and we prove that this bound is met only by $\Lambda _{23}^{*}$, the best known lattice covering in $\mathbb{R}^{23}$.

Bibliography

[1] A. Barvinok, A Course in Convexity. Graduate Studies in Mathematics 54, Amer. Math. Soc. 2002.  MR 1940576 |  Zbl 1014.52001
[2] H. Cohn, A. Kumar, Universally optimal distribution of points on spheres. J. Amer. Math. Soc. 20 (2007), 99–148.  MR 2257398 |  Zbl 1198.52009
[3] H. Cohn, A. Kumar, Optimality and uniqueness of the Leech lattice among lattices. Ann. of Math. 170 (2009), 1003–1050.  MR 2600869 |  Zbl 1213.11144
[4] J. H. Conway, R. A. Parker, N. J. A. Sloane, The covering radius of the Leech lattice. Proc. Roy. Soc. London Ser. A 380 (1982), 261–290.  Zbl 0496.10020
[5] J. H. Conway, N. J. A. Sloane, Sphere Packings, Lattices and Groups (third edition). Grundlehren der mathematischen Wissenschaften 290, Springer–Verlag, 1999.  MR 1662447 |  Zbl 0915.52003
[6] H. S. M. Coxeter, Extreme forms. Canadian J. Math. 3 (1951), 391–441.  MR 44580 |  Zbl 0044.04201
[7] P. Delsarte, J. M. Goethals, J. J. Seidel, Spherical codes and designs. Geometriae Dedicata 6 (1977), 363–388.  MR 485471 |  Zbl 0376.05015
[8] M. Deza, M. Dutour, The hypermetric cone on seven vertices. Experiment. Math. 12 (2004), 433–440.  MR 2043993 |  Zbl 1101.11021
[9] M. Deza, V. P. Grishukhin, M. Laurent, Extreme hypermetrics and L-polytopes. In G. Halász et al. eds, Sets, Graphs and Numbers, Budapest (Hungary), 1991, 60 Colloquia Mathematica Societatis János Bolyai (1992), 157–209.  MR 1218190 |  Zbl 0784.11027
[10] M. Deza, M. Laurent, Geometry of cuts and metrics. Springer–Verlag, 1997.  MR 1460488 |  Zbl 1210.52001
[11] M. Dutour, Infinite serie of extreme Delaunay polytopes. European J. Combin. 26 (2005), 129–132.  MR 2101040 |  Zbl 1062.52013
[12] M. Dutour Sikirić, A. Schürmann, F. Vallentin, Complexity and algorithms for computing Voronoi cells of lattices. Math. Comp. 78 (2009), 1713–1731.  MR 2501071 |  Zbl 1215.11067
[13] M. Dutour Sikirić, V. Grishukhin, How to compute the rank of a Delaunay polytope. European J. Combin. 28 (2007) 762–773.  MR 2300757 |  Zbl 1117.52015
[14] M. Dutour Sikirić, K. Rybnikov, Perfect but not generating Delaunay polytopes. Special issue of Symmetry Culture and Science on tesselations, Part II, 317–326.
[15] M. Dutour Sikirić, R. Erdahl, K. Rybnikov, Perfect Delaunay polytopes in low dimensions. Integers 7 (2007) A39.  MR 2342197 |  Zbl 1194.52018
[16] M. Dutour Sikirić, A. Schürmann, F. Vallentin, Inhomogeneous extreme forms. Ann. Inst. Fourier 62 (2012), 2227–2255. Cedram |  MR 3060757
[17] M. Dutour Sikirić, Enumeration of inhomogeneous perfect forms. In preparation.
[18] M. Dutour Sikirić, K. Rybnikov, A new algorithm in geometry of numbers. In Proceedings of ISVD-07, the 4-th International Symposium on Voronoi Diagrams in Science and Engineering, Pontypridd, Wales, July 2007. IEEE Publishing Services, 2007.
[19] M. Dutour Sikirić, A. Schürmann, F. Vallentin, The contact polytope of the Leech lattice. Discrete Comput. Geom. 44 (2010), 904–911.  MR 2728040 |  Zbl 1204.52015
[20] R. Erdahl, A convex set of second-order inhomogeneous polynomials with applications to quantum mechanical many body theory. Mathematical Preprint #1975-40, Queen’s University, Kingston, Ontario, 1975.
[21] R. Erdahl, A cone of inhomogeneous second-order polynomials. Discrete Comput. Geom. 8 (1992), 387–416.  MR 1176378 |  Zbl 0773.11042
[22] R. M. Erdahl, K. Rybnikov, Voronoi-Dickson Hypothesis on Perfect Forms and L-types. Peter Gruber Festshrift: Rendiconti del Circolo Matematiko di Palermo, Serie II, Tomo LII, part I (2002), 279–296.  Zbl 1116.52006
[23] R.M. Erdahl, A. Ordine, K. Rybnikov, Perfect Delaunay Polytopes and Perfect Quadratic Functions on Lattices. Integer points in polyhedra—geometry, number theory, representation theory, algebra, optimization, statistics, Contemporary Mathematics 452, American Mathematical Society, 2008, 93–114.  Zbl 1162.52006
[24] R. Erdahl, K. Rybnikov, An infinite series of perfect quadratic forms and big Delaunay simplices in $\mathbb{Z}^n$. Tr. Mat. Inst. Steklova 239 (2002), Diskret. Geom. i Geom. Chisel, 170–178; translation in Proc. Steklov Inst. Math. 239 (2002), 159–167.  Zbl 1126.11324
[25] V. Grishukhin, Infinite series of extreme Delaunay polytopes. European J. Combin. 27 (2006), 481–495.  MR 2215210 |  Zbl 1088.52010
[26] J.-M. Kantor, Lattice polytope: some open problems. AMS Snowbird Proceedings.
[27] P.W. Lemmens, J.J. Seidel, Equiangular lines. J. Algebra 24 (1973), 494–512.  MR 307969 |  Zbl 0255.50005
[28] J. Martinet, Perfect lattices in Euclidean spaces. Springer, 2003.  MR 1957723 |  Zbl 1017.11031
[29] R. E. O’Connor, G. Pall, The construction of integral quadratic forms of determinant $1$. Duke Math. J. 11 (1944), 319–331.  MR 10153 |  Zbl 0060.11103
[30] W. Plesken, B. Souvignier, Computing isometries of lattices. J. Symbolic Comput. 24 (1997), 327–334.  MR 1484483 |  Zbl 0882.11042
[31] W. Plesken, Finite unimodular groups of prime degree and circulants. J. of Algebra 97 (1985), 286–312.  MR 812182 |  Zbl 0583.20036
[32] A. Schürmann, Experimental study of energy-minimizing point configurations on spheres. Amer. Math. Soc. Univ. Lect. Ser. 2009.  Zbl 1185.68771
[33] W. Smith, PhD thesis: studies in computational geometry motivated by mesh generation. Department of Applied Mathematics, Princeton University, 1988.  MR 2637686
[34] B. B. Venkov, Réseaux et designs sphériques. In Réseaux euclidiens, designs sphériques et formes modulaires, edited by J. Martinet, Monographie numéro 37 de L’enseignement Mathématique, 2001.  MR 1878745 |  Zbl 1139.11320
[35] G. F. Voronoi, Nouvelles applications des paramètres continus à la théorie des formes quadratiques - Deuxième mémoire. J. für die Reine und Angewandte Mathematik, 134 (1908), 198-287 and 136 (1909), 67–178.  JFM 39.0274.01
[36] N. J. A. Sloane, G. Nebe, A Catalogue of Lattices. http://www2.research.att.com/~njas/lattices/. Software
[37] M. Dutour Sikirić, polyhedral, http://www.liga.ens.fr/~dutour/Polyhedral/.