2002
DOI: 10.1090/s0273-0979-02-00950-3
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From deep holes to free planes

Abstract: Abstract. During the last decades, by applying techniques from Number Theory, Combinatorics and Measure Theory, remarkable progress has been made in the study of deep holes, free planes and related topics in packings of convex bodies, especially in lattice ball packings. Meanwhile, some fascinating new problems have been proposed. To stimulate further research in related areas, we will review the main results, some key techniques and some fundamental problems about deep holes, free cylinders and free planes in… Show more

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Cited by 20 publications
(20 citation statements)
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“…N (Λ * ) = 1 and r(Λ * ) ≤ 4, say. Such lattices are well known to exists (see [61,42]). Then Theorem 5.1 implies that c 1 (R n /Λ) = Ω( √ n).…”
Section: Flat Tori Which Do Not Embed Into Lmentioning
confidence: 99%
“…N (Λ * ) = 1 and r(Λ * ) ≤ 4, say. Such lattices are well known to exists (see [61,42]). Then Theorem 5.1 implies that c 1 (R n /Λ) = Ω( √ n).…”
Section: Flat Tori Which Do Not Embed Into Lmentioning
confidence: 99%
“…In contrast to the covering case, the application of MAXDET to the triangulation constructed in Section 6 indicates that γ(E 8 ) = √ 2 may in fact be a local optimum, as conjectured by Zong [Zon02]. He conjectured that E 8 gives the global optimum.…”
Section: Remarks On the Packing-covering Problemmentioning
confidence: 91%
“…Its study has recently been given renewed attention by Zong [14,15,16,17,18] and others [6,12]. Important contributions to the non-lattice case have also been made by Linhardt [9], Böröczky [1], and Doyle, Lagarias and Randall [4].…”
Section: Introductionmentioning
confidence: 99%