Let $ C_{n} $ be an $ n $-dimensional cross-polytope and $ \Gamma_{p}(C_{n}) $ be the smallest positive number $ \gamma $ such that $ C_{n} $ can be covered by $ p $ translates of $ \gamma C_{n} $. We obtain better estimates of $ \Gamma_{2^n}(C_n) $ for small $ n $ and a universal upper bound of $ \Gamma_{2^n}(C_n) $ for all positive integers $ n $.
Citation: Feifei Chen, Shenghua Gao, Senlin Wu. Covering cross-polytopes with smaller homothetic copies[J]. AIMS Mathematics, 2024, 9(2): 4014-4020. doi: 10.3934/math.2024195
Let $ C_{n} $ be an $ n $-dimensional cross-polytope and $ \Gamma_{p}(C_{n}) $ be the smallest positive number $ \gamma $ such that $ C_{n} $ can be covered by $ p $ translates of $ \gamma C_{n} $. We obtain better estimates of $ \Gamma_{2^n}(C_n) $ for small $ n $ and a universal upper bound of $ \Gamma_{2^n}(C_n) $ for all positive integers $ n $.
| [1] |
K. Bezdek, Hadwiger's covering conjecture and its relatives, Am. Math. Mon., 99 (1992), 954–956. https://doi.org/10.1080/00029890.1992.11995963 doi: 10.1080/00029890.1992.11995963
|
| [2] | K. Bezdek, Classical topics in discrete geometry, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, New York: Springer, 2010. https://doi.org/10.1007/978-1-4419-0600-7 |
| [3] | K. Bezdek, M. A. Khan, The geometry of homothetic covering and illumination, In: Discrete Geometry and Symmetry, Cham: Springer, 2018. https://doi.org/10.1007/978-3-319-78434-2 |
| [4] | V. Boltyanski, H. Martini, P. S. Soltan, Excursions into combinatorial geometry, Universitext, Berlin: Springer-Verlag, 1997. https://doi.org/10.1007/978-3-642-59237-9 |
| [5] | P. Brass, W. Moser, J. Pach, Research problems in discrete geometry, New York: Springer, 2005. https://doi.org/10.1007/0-387-29929-7 |
| [6] |
H. Martini, V. Soltan, Combinatorial problems on the illumination of convex bodies, Aequationes Math., 57 (1999), 121–152. https://doi.org/10.1007/s000100050074 doi: 10.1007/s000100050074
|
| [7] |
F. W. Levi, Überdeckung eines Eibereiches durch Parallelverschiebungen seines offenen Kerns, Arch. Math., 6 (1955), 369–370. https://doi.org/10.1007/BF01900507 doi: 10.1007/BF01900507
|
| [8] |
C. Zong, A quantitative program for Hadwiger's covering conjecture, Sci. China Math., 53 (2010), 2551–2560. https://doi.org/10.1007/s11425-010-4087-3 doi: 10.1007/s11425-010-4087-3
|
| [9] |
X. Li, L. Meng, S. Wu, Covering functionals of convex polytopes with few vertices, Arch. Math., 119 (2022), 135–146. https://doi.org/10.1007/s00013-022-01727-z doi: 10.1007/s00013-022-01727-z
|
| [10] | F. Xue, Y. Lian, Y. Zhang, On Hadwiger's covering functional for the simplexand the cross-polytope, arXiv Preprint, 2021. https://doi.org/10.48550/arXiv.2108.13277 |
| [11] |
U. Betke, M. Henk, Intrinsic volumes and lattice points of crosspolytopes, Monatsh. Math., 115 (1993), 27–33. https://doi.org/10.1007/BF01311208 doi: 10.1007/BF01311208
|
| [12] | G. Pólya, G. Szegő, Problems and theorems in analysis: Series, integral calculus, theory of functions, Classics in Mathematics, Berlin: Springer-Verlag, 1998. https://doi.org/10.1007/978-3-642-61905-2 |
Feifei Chen, Shenghua Gao, Senlin Wu. Covering cross-polytopes with smaller homothetic copies[J]. AIMS Mathematics, 2024, 9(2): 4014-4020. doi: 10.3934/math.2024195
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