Lower bound on the translative covering density of octahedra
-
Yiming Li
,
Miao Fu
Abstract
Based on Zong’s work [26] on translative packing densities of 3-dimensional convex bodies, we present a local method to estimate the density θt(C3) of the densest translative covering of an octahedron. As a consequence we prove that θt(C3) ≥ 1 + 6.6 × 10–8, which is the first non-trivial lower bound for this density.
Funding statement: This work has been supported by the National Natural Science Foundation of China (NSFC12226006, NSFC11921001, NSFC11801410 and NSFC11971346), the Natural Key Research and Development Program of China (2018YFA0704701), the China Scholarship Council (No. 201906255029) and the Scientific Research Fund of Zhejiang Provincial Education Department (Y202250092), Y202351675) and the Natural Science Foundation of Zhejiang Provincial (LQ24A010008).
Acknowledgements
The authors are grateful to C. Zong for his supervision and discussion.
Communicated by: M. Henk
References
[1] R. P. Bambah, On lattice coverings by spheres. Proc. Nat. Inst. Sci. India 20 (1954), 25–52. MR61137 Zbl 0059.16301Suche in Google Scholar
[2] E. S. Barnes, The covering of space by spheres. Canadian J. Math. 8 (1956), 293–304. MR77576 Zbl 0072.0360310.4153/CJM-1956-033-4Suche in Google Scholar
[3] U. Betke, M. Henk, Densest lattice packings of 3-polytopes. Comput. Geom. 16 (2000), 157–186. MR1765181 Zbl 1133.5230710.1016/S0925-7721(00)00007-9Suche in Google Scholar
[4] P. Brass, W. Moser, J. Pach, Research problems in discrete geometry. Springer 2005. MR2163782 Zbl 1086.52001Suche in Google Scholar
[5] J. H. Conway, S. Torquato, Packing, tiling, and covering with tetrahedra. Proc. Natl. Acad. Sci. USA 103 (2006), 10612–10617. MR2242647 Zbl 1160.5230110.1073/pnas.0601389103Suche in Google Scholar PubMed PubMed Central
[6] B. N. Delone, S. S. Ryškov, Solution of the problem on the least dense lattice covering of a 4-dimensional space by equal spheres (Russian). Dokl. Akad. Nauk SSSR 152 (1963), 523–524. English translation: Soviet Math. Dokl. 4 (1963), 1333–1334. MR175850 Zbl 0132.03204Suche in Google 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 1056.0504610.1137/S0895480100372899Suche in Google Scholar
[8] I. Fáry, Sur la densité des réseaux de domaines convexes. Bull. Soc. Math. France 78 (1950), 152–161. MR39288 Zbl 0039.1820210.24033/bsmf.1413Suche in Google Scholar
[9] E. S. Fedorov, Elements of the study of figures (Russian). Zap. Mineral. Imper. S. Petersburgskogo Obs̆c̆. 21 (1885), 1–279.10.1038/scientificamerican11011885-21cbuildSuche in Google Scholar
[10] 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 0789.5201810.1016/B978-0-444-89597-4.50007-XSuche in Google Scholar
[11] L. Few, Covering space by spheres. Mathematika 3 (1956), 136–139. MR83525 Zbl 0072.2730210.1112/S0025579300001819Suche in Google Scholar
[12] 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 0916.0503310.1137/S0895480195286456Suche in Google Scholar
[13] R. Forcade, J. Lamoreaux, Lattice-simplex coverings and the 84-shape. SIAM J. Discrete Math. 13 (2000), 194–201. MR1760337 Zbl 0941.0502210.1137/S0895480198349622Suche in Google Scholar
[14] M. Fu, F. Xue, C. Zong, Lower bounds on lattice covering densities of simplices. SIAM J. Discrete Math. 37 (2023), 1788–1804. MR4625894 Zbl 0773487010.1137/22M1514155Suche in Google Scholar
[15] S. W. Golomb, L. R. Welch, Perfect codes in the Lee metric and the packing of polyominoes. SIAM J. Appl. Math. 18 (1970), 302–317. MR256766 Zbl 0192.5630210.1137/0118025Suche in Google Scholar
[16] J. Januszewski, Covering the plane with translates of a triangle. Discrete Comput. Geom. 43 (2010), 167–178. MR2575324 Zbl 1189.5201910.1007/s00454-009-9203-1Suche in Google Scholar
[17] R. Kershner, The number of circles covering a set. Amer. J. Math. 61 (1939), 665–671. MR43 Zbl 0021.1140110.2307/2371320Suche in Google Scholar
[18] J. C. Lagarias, C. Zong, Mysteries in packing regular tetrahedra. Notices Amer. Math. Soc. 59 (2012), 1540–1549. MR3027108 Zbl 1284.5201810.1090/noti918Suche in Google Scholar
[19] Y. Li, M. Fu, Y. Zhang, Lower Bound on Translative Covering Density of Tetrahedra. Discrete Comput. Geom., to appear. https://doi.org/10.1007/s00454-023-00602-010.1007/s00454-023-00602-0Suche in Google Scholar
[20] H. Minkowski, Dichteste gitterförmige Lagerung kongruenter Körper. Gött. Nachr. (1904), 311–355. Zbl 35.0508.02Suche in Google Scholar
[21] O. Ordentlich, O. Regev, B. Weiss, New bounds on the density of lattice coverings. J. Amer. Math. Soc. 35 (2022), 295–308. MR4322394 Zbl 1480.1108610.1090/jams/984Suche in Google Scholar
[22] C. A. Rogers, Packing and covering. Cambridge Univ. Press 1964. MR172183 Zbl 0176.51401Suche in Google Scholar
[23] S. S. Ryškov, E. P. Baranovskiĭ, Solution of the problem of the least dense lattice covering of five-dimensional space by equal spheres (Russian). Dokl. Akad. Nauk SSSR 222 (1975), 39–42. English translation: Soviet Math. Dokl. 16 (1975), no. 3, 586–590. MR427238 Zbl 0326.10026Suche in Google Scholar
[24] F. Xue, C. Zong, On lattice coverings by simplices. Adv. Geom. 18 (2018), 181–186. MR3785419 Zbl 1429.5202210.1515/advgeom-2017-0049Suche in Google Scholar
[25] C. Zong, Sphere packings. Springer 1999. MR1707318 Zbl 0935.52016Suche in Google Scholar
[26] C. Zong, On the translative packing densities of tetrahedra and cuboctahedra. Adv. Math. 260 (2014), 130–190. MR3209351 Zbl 1295.5202310.1016/j.aim.2014.04.009Suche in Google Scholar
[27] C. Zong, Packing, covering and tiling in two-dimensional spaces. Expo. Math. 32 (2014), 297–364. MR3279483 Zbl 1307.5201210.1016/j.exmath.2013.12.002Suche in Google Scholar
© 2024 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Inequalities for f*-vectors of lattice polytopes
- Lower bound on the translative covering density of octahedra
- Variations on the Weak Bounded Negativity Conjecture
- Poisson Structures on moduli spaces of Higgs bundles over stacky curves
- Generalized Shioda–Inose structures of order 3
- Deformation cones of Tesler polytopes
- Some observations on conformal symmetries of G2-structures
- Characterization of the sphere and of bodies of revolution by means of Larman points
- Fractional-linear integrals of geodesic flows on surfaces and Nakai’s geodesic 4-webs
- The feet of orthogonal Buekenhout–Metz unitals
Artikel in diesem Heft
- Frontmatter
- Inequalities for f*-vectors of lattice polytopes
- Lower bound on the translative covering density of octahedra
- Variations on the Weak Bounded Negativity Conjecture
- Poisson Structures on moduli spaces of Higgs bundles over stacky curves
- Generalized Shioda–Inose structures of order 3
- Deformation cones of Tesler polytopes
- Some observations on conformal symmetries of G2-structures
- Characterization of the sphere and of bodies of revolution by means of Larman points
- Fractional-linear integrals of geodesic flows on surfaces and Nakai’s geodesic 4-webs
- The feet of orthogonal Buekenhout–Metz unitals
