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Lattice packing and covering of convex bodies

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Abstract

The aim of this article is twofold. First, to indicate briefly major problems and developments dealing with lattice packings and coverings of balls and convex bodies. Second, to survey more recent results on uniqueness of lattice packings and coverings of extreme density, on characterization of local minima and maxima of the density and on estimates of the kissing number. Emphasis is on results in general dimensions.

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References

  1. K. Ball, “A Lower Bound for the Optimal Density of Lattice Packings,” Int. Math. Res. Not., No. 10, 217–221 (1992).

  2. R. P. Bambah, “On Lattice Coverings by Spheres,” Proc. Natl. Inst. Sci. India 20, 25–52 (1954).

    MathSciNet  MATH  Google Scholar 

  3. E. S. Barnes, “On a Theorem of Voronoĭ,” Proc. Cambridge Philos. Soc. 53, 537–539 (1957).

    Article  MathSciNet  MATH  Google Scholar 

  4. E. S. Barnes and T. J. Dickson, “Extreme Coverings of n-Space by Spheres,” J. Aust. Math. Soc. 7, 115–127 (1967).

    Article  MathSciNet  Google Scholar 

  5. A.-M. Bergé and J. Martinet, “Sur un problème de dualité lié aux sphères en géométrie des nombres,” J. Number Theory 32, 14–42 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  6. A.-M. Bergé and J. Martinet, “Sur la classification des réseaux eutactiques,” J. London Math. Soc., Ser. 2,53, 417–432 (1996).

    Google Scholar 

  7. U. Betke and M. Henk, “Densest Lattice Packings of 3-Polytopes,” Comput. Geom. 16, 157–186 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  8. H. F. Blichfeldt, “The Minimum Value of Quadratic Forms, and the Closest Packing of Spheres,” Math. Ann. 101, 605–608 (1929).

    Article  MathSciNet  MATH  Google Scholar 

  9. H. F. Blichfeldt, “The Minimum Value of Positive Quadratic Forms in Six, Seven and Eight Variables,” Math. Z. 39, 1–15 (1934).

    Article  MathSciNet  Google Scholar 

  10. H. Cohn and A. Kumar, “The Densest Lattice in Twenty-Four Dimensions,” Electron. Res. Announc. Am. Math. Soc. 10, 58–67 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  11. J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups. With additional contributions by E. Bannai, R. E. Borcherds, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen, and B. B. Venkov, 3rd ed. (Springer, New York, 1999), Grundl. Math. Wissensch. 290.

    MATH  Google Scholar 

  12. R. Coulangeon, “Spherical Designs and Zeta Functions of Lattices,” Int. Math. Res. Not., No. 25, 49620 (2006).

  13. B. Delaunay, “Sur la sphère vide,” in Proc. Int. Math. Congr., Toronto, 1924 (Univ. Toronto Press, Toronto, 1928), Vol. 1, pp. 695–700.

    Google Scholar 

  14. B. N. Delone, N. P. Dolbilin, S. S. Ryškov, and M. I. Štogrin, “A New Construction in the Theory of Lattice Coverings of an n-Dimensional Space by Equal Spheres,” Izv. Akad. Nauk SSSR, Ser. Mat. 34(2), 289–298 (1970) [Math. USSR, Izv. 4 (2), 293–302 (1970)].

    MathSciNet  MATH  Google Scholar 

  15. B. N. Delone and S. S. Ryshkov, “Solution of the Problem of the Least Dense Lattice Covering of a Four-Dimensional Space by Equal Spheres,” Dokl. Akad. Nauk SSSR 152(3), 523–524 (1963) [Sov. Math., Dokl. 4, 1333–1334 (1964)].

    MathSciNet  Google Scholar 

  16. B. N. Delone and S. S. Ryshkov, “A Contribution to the Theory of the Extrema of a Multidimensional ζ-Function,” Dokl. Akad. Nauk SSSR 173(4), 991–994 (1967) [Sov. Math., Dokl. 8, 499–503 (1967)].

    Google Scholar 

  17. M. Dutour Sikirić, A. Schürmann, and F. Vallentin, “Classification of Eight-Dimensional Perfect Forms,” Electron. Res. Announc. Am. Math. Soc. 13, 21–32 (2007).

    Article  MATH  Google Scholar 

  18. P. Erdös, P. M. Gruber, and J. Hammer, Lattice Points (Longman Scientific & Technical, Harlow, 1989).

    MATH  Google Scholar 

  19. P. Erdös and C. A. Rogers, “The Star Number of Coverings of Space with Convex Bodies,” Acta Arith. 9, 41–45 (1964).

    MathSciNet  MATH  Google Scholar 

  20. L. Fejes Tòth, Lagerungen in der Ebene, auf der Kugel und im Raum, 2nd ed. (Springer, Berlin, 1972), Grundl. Math. Wissensch. 65.

    MATH  Google Scholar 

  21. C. F. Gauss, “Recension der’ Untersuchungen über die Eigenschaften der positiven ternären quadratischen Formen von Ludwig August Seeber, 1831’,” J. Reine Angew. Math. 20, 312–320 (1840); in Werke (Kön. Ges. Wissensch., Göttingen, 1863), Vol. 2, pp. 188–196.

    Google Scholar 

  22. P. M. Gruber, “Typical Convex Bodies Have Surprisingly Few Neighbours in Densest Lattice Packings,” Stud. Sci. Math. Hung. 21, 163–173 (1986).

    MATH  Google Scholar 

  23. P. M. Gruber, “Minimal Ellipsoids and Their Duals,” Rend. Circ. Mat. Palermo, Ser. 2,37, 35–64 (1988).

    Article  Google Scholar 

  24. P. M. Gruber, “Baire Categories in Convexity,” in Handbook of Convex Geometry (North-Holland, Amsterdam, 1993), Vol. B, pp.1327–1346.

    Google Scholar 

  25. P. M. Gruber, “In Many Cases Optimal Configurations Are Almost Regular Hexagonal,” Suppl. Rend. Circ. Mat. Palermo, Ser. 2, 65, 121–145 (2000).

    Google Scholar 

  26. P. M. Gruber, Convex and Discrete Geometry (Springer, Berlin, 2007), Grundl. Math. Wissensch. 336.

    MATH  Google Scholar 

  27. P. M. Gruber, “Application of an Idea of Voronoĭ to John Type Problems,” Adv. Math. 218, 309–351 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  28. P. M. Gruber, “Application of an Idea of Voronoĭ, a Report” (in press).

  29. P. M. Gruber, “Uniqueness of Lattice Packings and Coverings of Extreme Density,” Adv. Geom. 11(4), 691–710 (2011).

    Article  MATH  Google Scholar 

  30. P. M. Gruber, “Application of an Idea of Voronoĭ to Lattice Packing” (submitted).

  31. P. M. Gruber, “Application of an Idea of Voronoĭ to Lattice Zeta Functions,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 276 (2012) (in press).

  32. P. M. Gruber, “Voronoĭ Type Criteria for Lattice Coverings with Balls,” Acta Arith. 149, 371–381 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  33. P. M. Gruber, “John and Loewner Ellipsoids,” Discrete Comput. Geom. 46(4), 776–788 (2011).

    Article  MATH  Google Scholar 

  34. P. M. Gruber and C. G. Lekkerkerker, Geometry of Numbers, 2nd ed. (North-Holland, Amsterdam, 1987; Nauka, Moscow, 2008).

    MATH  Google Scholar 

  35. P. M. Gruber and F. E. Schuster, “An Arithmetic Proof of John’s Ellipsoid Theorem,” Arch. Math. 85, 82–88 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  36. E. Hlawka, “Zur Geometrie der Zahlen,” Math. Z. 49, 285–312 (1943).

    Article  MathSciNet  MATH  Google Scholar 

  37. G. A. Kabatyanskiĭ and V. I. Levenshtein, “On Bounds for Packings on a Sphere and in Space,” Probl. Peredachi Inf. 14(1), 3–25 (1978) [Probl. Inf. Transm. 14, 1–17 (1978)].

    Google Scholar 

  38. J. Kepler, Strena seu de nive sexangula (Tampach, Frankfurt, 1611); Gesammelte Werke (C.H. Beck’sche Verlag., München, 1940), Vol. 4; The Six-Cornered Snowflake (Clarendon Press, Oxford, 1966).

    Google Scholar 

  39. R. Kershner, “The Number of Circles Covering a Set,” Am. J. Math. 61, 665–671 (1939).

    Article  MathSciNet  Google Scholar 

  40. F. Klein, “Die allgemeine lineare Transformation der Liniencoordinaten,” Math. Ann. 2, 366–370 (1870); in Gesammelte mathematische Abhandlungen (Springer, Berlin, 1921), Vol. 1, pp. 81–86.

    Article  MathSciNet  Google Scholar 

  41. M. Kneser, “Two Remarks on Extreme Forms,” Can. J. Math. 7, 145–149 (1955).

    Article  MathSciNet  MATH  Google Scholar 

  42. A. Korkine and G. Zolotareff, “Sur les formes quadratiques positives quaternaires,” Math. Ann. 5, 581–583 (1872).

    Article  MathSciNet  Google Scholar 

  43. A. Korkine and G. Zolotareff, “Sur les formes quadratiques,” Math. Ann. 6, 366–389 (1873).

    Article  MathSciNet  Google Scholar 

  44. A. Korkine and G. Zolotareff, “Sur les formes quadratiques positives,” Math. Ann. 11, 242–292 (1877).

    Article  MathSciNet  Google Scholar 

  45. J. Leech and N. J. A. Sloane, “Sphere Packings and Error-Correcting Codes,” Can. J. Math. 23, 718–745 (1971).

    Article  MathSciNet  MATH  Google Scholar 

  46. J. Martinet, Perfect Lattices in Euclidean Spaces (Springer, Berlin, 2003), Grundl. Math. Wissensch. 325.

    MATH  Google Scholar 

  47. H. Minkowski, Diophantische Approximationen: Eine Einführung in die Zahlentheorie (Teubner, Leipzig, 1907, 1927; Chelsea, New York, 1957; Physica-Verlag, Würzburg, 1961).

    MATH  Google Scholar 

  48. F. Morgan and R. Bolton, “Hexagonal Economic Regions Solve the Location Problem,” Am. Math. Mon. 109, 165–172 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  49. J. Plücker, Neue Geometrie des Raumes gegründet auf die Betrachtung der geraden Linie als Raumelement, I. Abth. 1868 mit einem Vorwort von A. Clebsch; II. Abth. 1869, herausgegeben von F. Klein (Teubner, Leipzig, 1868).

  50. C. A. Rogers, Packing and Covering (Cambridge Univ. Press, Cambridge, 1964).

    MATH  Google Scholar 

  51. J. A. Rush, “A Lower Bound on Packing Density,” Invent. Math. 98, 499–509 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  52. S. S. Ryshkov, “On the Question of Final ζ-Optimality of Lattices Providing the Closest Lattice Packing of n-Dimensional Spheres,” Sib. Mat. Zh. 14(5), 1065–1075 (1973) [Sib. Math. J. 14, 743–750 (1974)].

    MATH  Google Scholar 

  53. S. S. Ryshkov, “Geometry of Positive Quadratic Forms,” in Proc. Int. Congr. Math., Vancouver, 1974 (Can. Math. Congr., Montreal, 1975), Vol. 1, pp. 501–506 [in Russian].

    Google Scholar 

  54. S. S. Ryshkov and E. P. Baranovskii, “Solution of the Problem of Least Dense Lattice Covering of Five-Dimensional Space by Equal Spheres,” Dokl. Akad. Nauk SSSR 222(1), 39–42 (1975) [Sov. Math., Dokl. 16, 586–590 (1975)].

    MathSciNet  Google Scholar 

  55. S. S. Ryshkov and E. P. Baranovskii, “Classical Methods in the Theory of Lattice Packings,” Usp. Mat. Nauk 34(4), 3–63 (1979) [Russ. Math. Surv. 34 (4), 1–68 (1979)].

    MathSciNet  MATH  Google Scholar 

  56. E. B. Saff and A. B. J. Kuijlaars, “Distributing Many Points on a Sphere,” Math. Intell. 19(1), 5–11 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  57. P. Sarnak and A. Strömbergsson, “Minima of Epstein’s Zeta Function and Heights of Flat Tori,” Invent. Math. 165, 115–151 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  58. W. M. Schmidt, “On the Minkowski-Hlawka Theorem,” Ill. J. Math. 7, 18–23 (1963).

    Google Scholar 

  59. P. Schmutz, “Riemann Surfaces with Shortest Geodesic of Maximal Length,” Geom. Funct. Anal. 3, 564–631 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  60. P. Schmutz, “Systoles on Riemann Surfaces,” Manuscr. Math. 85, 429–447 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  61. P. Schmutz Schaller, “Geometry of Riemann Surfaces Based on Closed Geodesics,” Bull. Am. Math. Soc. 35, 193–214 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  62. P. Schmutz Schaller, “Perfect Non-extremal Riemann Surfaces,” Can. Math. Bull. 43, 115–125 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  63. A. Schürmann, Computational Geometry of Positive Definite Quadratic Forms: Polyhedral Reduction Theories, Algorithms, and Applications (Am. Math. Soc., Providence, RI, 2009).

    MATH  Google Scholar 

  64. A. Schürmann and F. Vallentin, “Computational Approaches to Lattice Packing and Covering Problems,” Discrete Comput. Geom. 35, 73–111 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  65. V. M. Sidel’nikov, “On the Densest Packing of Balls on the Surface of an n-Dimensional Euclidean Sphere and the Number of Binary Code Vectors with a Given Code Distance,” Dokl. Akad. Nauk SSSR 213(5), 1029–1032 (1973) [Sov. Math., Dokl. 14, 1851–1855 (1973)].

    MathSciNet  Google Scholar 

  66. C. L. Siegel, Lectures on the Geometry of Numbers (Springer, Berlin, 1989).

    MATH  Google Scholar 

  67. H. P. F. Swinnerton-Dyer, “Extremal Lattices of Convex Bodies,” Proc. Cambridge Philos. Soc. 49, 161–162 (1953).

    Article  MathSciNet  MATH  Google Scholar 

  68. B. Venkov, “Réseaux et designs sphériques,” in Réseaux euclidiens, designs sphériques et formes modulaires (Enseign. Math., Geneva, 2001), Monogr. Enseign. Math. 37, pp. 10–86.

    Google Scholar 

  69. G. Voronoï, “Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Premièr mémoire: Sur quelques propriétés des formes quadratiques positives parfaites,” J. Reine Angew. Math. 133, 97–178 (1908). See Russ. transl. in [71, Vol. 2, pp. 171–238].

    Article  MATH  Google Scholar 

  70. G. Voronoï, “Nouvelles applications des paramètres continus à la théorie des formes quadratiques. Deuxième mémoire: Recherches sur les paralléloèdres primitifs. I, II,” J. Reine Angew. Math. 134, 198–287 (1908); 136, 67–181 (1909). See Russ. transl. in [71, Vol. 2, pp. 239–368].

    Article  MATH  Google Scholar 

  71. G. F. Voronoĭ, Collected Works (Izd. Akad. Nauk Ukr. SSR, Kiev, 1952, 1953), Vols. 1–3 [in Russian].

    MATH  Google Scholar 

  72. Voronoï’s Impact on Modern Science, Ed. by P. Engel and H. Syta (Inst. Math. Natl. Acad. Sci. Ukraine, Kyiv, 1998), Books 1, 2.

    Google Scholar 

  73. T. Watanabe, “A Survey on Voronoï’s Theorem,” in Geometry and Analysis of Automorphic Forms of Several Variables: Proc. Int. Symp., Tokyo, Sept. 14–17, 2009 (World Sci., Singapore, 2011), p. 334.

    Google Scholar 

  74. C. Zong, Sphere Packings (Springer, New York, 1999).

    MATH  Google Scholar 

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  1. Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstraße 8-10/104, A-1040, Vienna, Austria

    Peter M. Gruber

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Gruber, P.M. Lattice packing and covering of convex bodies. Proc. Steklov Inst. Math. 275, 229–238 (2011). https://doi.org/10.1134/S0081543811080165

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