The simultaneous lattice packing-covering constant of octahedra
-
Yiming Li
Abstract
This paper proves that the simultaneous lattice packing-covering constant of an octahedron is 7/6. In other words, 7/6 is the smallest positive number r such that for every octahedron O centered at the origin there is a lattice Λ such that O + Λ is a packing in 𝔼3 and rO + Λ is a covering of 𝔼3.
Funding statement: This work is supported by the National Natural Science Foundation of China (NSFC12226006, NSFC11921001) and the Natural Key Research and Development Program of China (2018YFA0704701).
Communicated by: F. Santos
References
[1] U. Betke, M. Henk, Densest lattice packings of 3-polytopes. Comput. Geom. 16 (2000), 157–186. MR1765181 Zbl 1133.5230710.1016/S0925-7721(00)00007-9Search in Google Scholar
[2] K. Böröczky, Closest packing and loosest covering of the space with balls. Studia Sci. Math. Hungar. 21 (1986), 79–89. MR898846 Zbl 0639.52013Search in Google Scholar
[3] P. Brass, W. Moser, J. Pach, Research problems in discrete geometry. Springer 2005. MR2163782 Zbl 1086.52001Search in Google Scholar
[4] G. J. Butler, Simultaneous packing and covering in euclidean space. Proc. London Math. Soc. (3) 25 (1972), 721–735. MR319054 Zbl 0251.5202210.1112/plms/s3-25.4.721Search 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.0601389103Search in Google Scholar PubMed PubMed Central
[6] L. Fejes Tóth, Close packing and loose covering with balls. Publ. Math. Debrecen 23 (1976), 323–326. MR428199 Zbl 0357.5201010.5486/PMD.1976.23.3-4.21Search in Google Scholar
[7] T. C. Hales, On the Reinhardt conjecture. Vietnam J. Math. 39 (2011), 287–307. MR2919755 Zbl 1239.52002Search in Google Scholar
[8] M. Henk, Finite and Infinite Packings. Habilitationsschrift, Universität Siegen, 1995.Search in Google Scholar
[9] M. Henk, A note on lattice packings via lattice refinements. Exp. Math. 27 (2018), 1–9. MR3750923 Zbl 1388.5201110.1080/10586458.2016.1208595Search in Google Scholar
[10] J. Horváth, On close lattice packing of unit spheres in the space En Proc. Steklov Math. Inst. 152 (1982), 237–254. Translation from Trudy Mat. Inst. Steklova 152 (1980), 216–231. Zbl 0533.52009Search in Google Scholar
[11] J. Horváth, Several problems of n-dimensional discrete geometry (Russian). Doctoral dissertation, Steklov Math. Inst., Moskow, 1986.Search in Google Scholar
[12] J. Linhart, Closest packings and closest coverings by translates of a convex disc. Studia Sci. Math. Hungar. 13 (1978), 157–162 (1981). MR630388 Zbl 0417.52012Search in Google Scholar
[13] D. Micciancio, Almost perfect lattices, the covering radius problem, and applications to Ajtai’s connection factor. SIAM J. Comput. 34 (2004), 118–169. MR2114308 Zbl 1112.6806710.1137/S0097539703433511Search in Google Scholar
[14] H. Minkowski, Dichteste gitterförmige Lagerung kongruenter Körper. Nachr. K. Ges. Wiss. Göttingen (1904), 311–355. JFM 35.0508.02Search in Google Scholar
[15] C. A. Rogers, A note on coverings and packings. J. London Math. Soc. 25 (1950), 327–331. MR43819 Zbl 0038.0290210.1112/jlms/s1-25.4.327Search in Google Scholar
[16] C. A. Rogers, Lattice covering of space: The Minkowski–Hlawka theorem. Proc. London Math. Soc. (3) 8 (1958), 447–465. MR96639 Zbl 0083.2630210.1112/plms/s3-8.3.447Search in Google Scholar
[17] C. L. Siegel, A mean value theorem in geometry of numbers. Ann. of Math. (2) 46 (1945), 340–347. MR12093 Zbl 0063.0701110.2307/1969027Search in Google Scholar
[18] L. F. Tóth, Remarks on the closest packing of convex discs. Comment. Math. Helv. 53 (1978), 536–541. MR514025 Zbl 0406.5200610.1007/BF02566097Search in Google Scholar
[19] C. Zong, Sphere packings. Springer 1999. MR1707318 Zbl 0935.52016Search in Google Scholar
[20] C. Zong, From deep holes to free planes. Bull. Amer. Math. Soc. (N.S.) 39 (2002), 533–555. MR1920280 Zbl 1032.5200910.1090/S0273-0979-02-00950-3Search in Google Scholar
[21] C. Zong, Simultaneous packing and covering in the Euclidean plane. Monatsh. Math. 134 (2002), 247–255. MR1883504 Zbl 1003.1103110.1007/s605-002-8260-3Search in Google Scholar
[22] C. Zong, Simultaneous packing and covering in three-dimensional Euclidean space. J. London Math. Soc. (2) 67 (2003), 29–40. MR1942409 Zbl 1119.1103710.1112/S0024610702003873Search in Google Scholar
[23] C. Zong, The simultaneous packing and covering constants in the plane. Adv. Math. 218 (2008), 653–672. MR2414317 Zbl 1141.5202310.1016/j.aim.2008.01.007Search in Google Scholar
© 2025 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Polyhedral combinatorics of bisectors
- Explicit construction of decomposable Jacobians
- Some rigidity results on q-solitons
- Quasimorphisms on nonorientable surface diffeomorphism groups
- On nearly optimal paper Moebius bands
- Desargues’ Theorem in Laguerre Planes
- Simple obstructions and cone reduction
- The Bogomolov–Gieseker–Koseki inequality on surfaces with canonical singularities in arbitrary characteristic
- The simultaneous lattice packing-covering constant of octahedra
- An Abel–Jacobi theorem for metrized complexes of Riemann surfaces
- Exploring the interplay of semistable vector bundles and their restrictions on reducible curves
- Another proof of Segre’s theorem about ovals
Articles in the same Issue
- Frontmatter
- Polyhedral combinatorics of bisectors
- Explicit construction of decomposable Jacobians
- Some rigidity results on q-solitons
- Quasimorphisms on nonorientable surface diffeomorphism groups
- On nearly optimal paper Moebius bands
- Desargues’ Theorem in Laguerre Planes
- Simple obstructions and cone reduction
- The Bogomolov–Gieseker–Koseki inequality on surfaces with canonical singularities in arbitrary characteristic
- The simultaneous lattice packing-covering constant of octahedra
- An Abel–Jacobi theorem for metrized complexes of Riemann surfaces
- Exploring the interplay of semistable vector bundles and their restrictions on reducible curves
- Another proof of Segre’s theorem about ovals
