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Advances in Geometry

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Volume 18, Issue 2

Issues

On lattice coverings by simplices

F. Xue / C. Zong
  • Corresponding author
  • School of Mathematical Sciences, Peking University, Beijing 100871, China
  • Center for Applied Mathematics, Tianjin University, Tianjin 300072, China
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Published Online: 2018-01-07 | DOI: https://doi.org/10.1515/advgeom-2017-0049

Abstract

By studying the volumes of generalized difference bodies, this paper presents the first nontrivial lower bound for the lattice covering density by n-dimensional simplices.

Keywords: Lattice covering; simplex

MSC 2010: 52C17; 52B10; 52C07

References

  • [1]

    S. Artstein-Avidan, K. Einhorn, D. I. Florentin, Y. Ostrover, On Godbersen’s conjecture. Geom. Dedicata 178 (2015), 337–350. MR3397498 Zbl 1335.52005CrossrefGoogle Scholar

  • [2]

    R. P. Bambah, On lattice coverings by spheres. Proc. Nat. Inst. Sci. India 20 (1954), 25–52. MR0061137 Zbl 0059.16301Google Scholar

  • [3]

    E. S. Barnes, The covering of space by spheres. Canad. J. Math. 8 (1956), 293–304. MR0077576 Zbl 0072.03603CrossrefGoogle Scholar

  • [4]

    U. Betke, M. Henk, Densest lattice packings of 3-polytopes. Comput. Geom. 16 (2000), 157–186. MR1765181 Zbl 1133.52307CrossrefGoogle Scholar

  • [5]

    P. Brass, W. Moser, J. Pach, Research problems in discrete geometry. Springer 2005. MR2163782 Zbl 1086.52001Google Scholar

  • [6]

    J. H. Conway, S. Torquato, Packing, tiling, and covering with tetrahedra. Proc. Natl. Acad. Sci. USA 103 (2006), 10612–10617. MR2242647 Zbl 1160.52301CrossrefGoogle Scholar

  • [7]

    R. Dougherty, V. Faber, The degree-diameter problem for several varieties of Cayley graphs. I. The abelian case. SIAM J. Discrete Math. 17 (2004), 478–519. MR2050686 Zbl 1056.05046CrossrefGoogle Scholar

  • [8]

    I. Fáry, Sur la densité des réseaux de domaines convexes. Bull. Soc. Math. France 78 (1950), 152–161. MR0039288 Zbl 0039.18202Google Scholar

  • [9]

    E. S. Fedorov, Elements of the study of figures (Russian). Zap. Mineral. Imper. S. Petersburgskogo Obšč. (2) 21 (1885), 1–297. Also Izdat. Akad. Nauk SSSR, Moscow 1953. MR0062061Google Scholar

  • [10]

    L. Fejes, Eine Bemerkung über die Bedeckung der Ebene durch Eibereiche mit Mittelpunkt. Acta Univ. Szeged. Sect. Sci. Math. 11 (1946), 93–95. MR0017559 Zbl 0063.01334Google Scholar

  • [11]

    G. Fejes Tóth, W. Kuperberg, Packing and covering with convex sets. In: Handbook of convex geometry, Vol. A, B, 799–860, North-Holland 1993. MR1242997 Zbl 0789.52018Google Scholar

  • [12]

    L. Fejes Tóth, Some packing and covering theorems. Acta Sci. Math. Szeged 12 (1950), 62–67. MR0038086 Zbl 0037.22102Google Scholar

  • [13]

    L. Few, Covering space by spheres. Mathematika 3 (1956), 136–139. MR0083525 Zbl 0072.27302CrossrefGoogle Scholar

  • [14]

    C. M. Fiduccia, R. W. Forcade, J. S. Zito, Geometry and diameter bounds of directed Cayley graphs of abelian groups. SIAM J. Discrete Math. 11 (1998), 157–167. MR1612881 Zbl 0916.05033CrossrefGoogle Scholar

  • [15]

    R. Forcade, J. Lamoreaux, Lattice-simplex coverings and the 84-shape. SIAM J. Discrete Math. 13 (2000), 194–201. MR1760337 Zbl 0941.05022CrossrefGoogle Scholar

  • [16]

    C. Godbersen, Der Satz vom Vektorbereich in Räumen beliebiger Dimensionen. Dissertation, Göttingen, 1938.Google Scholar

  • [17]

    P. M. Gruber, C. G. Lekkerkerker, Geometry of numbers, volume 37 of North-Holland Mathematical Library. North-Holland 1987. MR893813 Zbl 0611.10017Google Scholar

  • [18]

    H. Hadwiger, Überdeckung des Raumes durch translationsgleiche Punktmengen und Nachbarnzahl. Monatsh. Math. 73 (1969), 213–217. MR0254745 Zbl 0182.55702CrossrefGoogle Scholar

  • [19]

    A. Hajnal, E. Makai, Research problems. Period. Math. Hungar. 5 (1974), 353–354. MR1553587CrossrefGoogle Scholar

  • [20]

    D. Hilbert, Mathematische Probleme. Arch. Math. Phys. 3 (1901), 44–63, 213–237. English translation: Bull. Amer. Math. Soc. 37 (2000), 407–436. MR1779412 Zbl 32.0084.05Google Scholar

  • [21]

    J. Januszewski, Covering the plane with translates of a triangle. Discrete Comput. Geom. 43 (2010), 167–178. MR2575324 Zbl 1189.52019CrossrefWeb of ScienceGoogle Scholar

  • [22]

    R. Kershner, The number of circles covering a set. Amer. J. Math. 61 (1939), 665–671. MR0000043 Zbl 0021.11401 JFM 65.0197.03CrossrefGoogle Scholar

  • [23]

    J. C. Lagarias, C. Zong, Mysteries in packing regular tetrahedra. Notices Amer. Math. Soc. 59 (2012), 1540–1549. MR3027108 Zbl 1284.52018Google Scholar

  • [24]

    H. Minkowski, Dichteste gitterförmige Lagerung kongruenter Körper. Nachr. K. Ges. Wiss. Göttingen, Math.-Phys. Kl. (1904), 311–355. Zbl 35.0508.02Google Scholar

  • [25]

    C. A. Rogers, Packing and covering. Cambridge Univ. Press 1964. MR0172183 Zbl 0176.51401Google Scholar

  • [26]

    C. A. Rogers, G. C. Shephard, The difference body of a convex body. Arch. Math. (Basel) 8 (1957), 220–233. MR0092172 Zbl 0082.15703CrossrefGoogle Scholar

  • [27]

    W. M. Schmidt, Zur Lagerung kongruenter Körper im Raum. Monatsh. Math. 65 (1961), 154–158. MR0126215 Zbl 0111.34905CrossrefGoogle Scholar

  • [28]

    R. Schneider, Convex bodies: the Brunn–Minkowski theory, volume 44 of Encyclopedia of Mathematics and its Applications. Cambridge Univ. Press 1993. MR1216521 Zbl 0798.52001Google Scholar

  • [29]

    C. Zong, Sphere packings. Springer 1999. MR1707318 Zbl 0935.52016Google Scholar

About the article


Received: 2016-03-06

Published Online: 2018-01-07

Published in Print: 2018-04-25


Funding: This work is supported by 973 Program 2013CB834201 and the Chang Jiang Scholars Program of China.


Citation Information: Advances in Geometry, Volume 18, Issue 2, Pages 181–186, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2017-0049.

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