Strange Phenomena in Convex and Discrete Geometry

Zong, Chuanming; Dudziak, James J.

doi:10.1007/978-1-4613-8481-6

Cited by 45 publications

(27 citation statements)

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“…It is well-known that for every n-dimensional centrally symmetric convex body C there is a linear transformation L such that [48]). Therefore, by Theorem 6.6 one can deduce…”

Section: Corollary 61 For Every Convex Body K Even If Not Centrallmentioning

confidence: 99%

From deep holes to free planes

Zong

2002

Bull. Amer. Math. Soc.

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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 this paper. Let E n denote the n-dimensional Euclidean space and let x 1 , x 2 denote the Euclidean distance between two points x 1 and x 2 . For convenience, we call the minimum distance between distinct points of a set its separation. Based on Question 0.1 it is natural to ask the following question. By a routine argument one can show that the minimum is 2/ √ 3 and every optimal set is isometric to Background and exampleswhere Z indicates the integer ring. In this case, one can triangulate E 2 into congruent regular triangles of edge length 2 and with all their vertices belonging to Λ 2 .

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“…It is well-known that for every n-dimensional centrally symmetric convex body C there is a linear transformation L such that [48]). Therefore, by Theorem 6.6 one can deduce…”

Section: Corollary 61 For Every Convex Body K Even If Not Centrallmentioning

confidence: 99%

From deep holes to free planes

Zong

2002

Bull. Amer. Math. Soc.

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“…"Most" means "all, except those in a set of the first Baire category". For other geometrically relevant, sometimes curious phenomena on convex surfaces, obtained via Baire categories, see [7], [23], [28]. Lemma 1.…”

Section: Nonrectifiable Compact Subsets Of the Cut Locusmentioning

confidence: 99%

Extreme points of the distance function on convex surfaces

Zamfirescu

1998

Trans. Amer. Math. Soc.

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Abstract. We first see that, in the sense of Baire categories, many convex surfaces have quite large cut loci and infinitely many relative maxima of the distance function from a point. Then we find that, on any convex surface, all these extreme points lie on a single subtree of the cut locus, with at most three endpoints. Finally, we confirm (both in the sense of measure and in the sense of Baire categories) Steinhaus' conjecture that "almost all" points admit a single farthest point on the surface.

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“…For additional related results and references on this topic, see [1], [2], [20], [22], and the survey papers [4] and [5].…”

Section: Moreover the Arrangement Of 18 Mutually Nonoverlapping Tranmentioning

confidence: 99%

“…We note that the method of "weight" functions, or "cost" functions, which are usually piecewise constant, is often used in combinatorics and discrete geometry. For some nice examples of this method, see [8] (see also [3]), [21], and [22]. Our method is a refinement of the one used by Zong [21].…”

Section: Notation and Terminologymentioning

confidence: 99%

The Translative Kissing Number of Tetrahedra Is 18

Talata

1999

Discrete Comput Geom

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We show that the maximum number of mutually nonoverlapping translates of any tetrahedron T which touch T is 18. Moreover, in the case of 18 touching translates the arrangement turns out to be unique. We also give a description of all possible arrangements of 17 touching translates. Finally, we apply these results to determine the minimum and maximum densities of 17 + -neighbor translative packings of tetrahedra.

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Strange Phenomena in Convex and Discrete Geometry

Cited by 45 publications

References 0 publications

From deep holes to free planes

From deep holes to free planes

Extreme points of the distance function on convex surfaces

The Translative Kissing Number of Tetrahedra Is 18

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