On contact numbers of totally separable unit sphere packings

Bezdek, Károly; Szalkai, Balázs; Szalkai, István

doi:10.1016/j.disc.2015.10.002

Cited by 14 publications

(37 citation statements)

References 15 publications

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“…The tools used in proving each statement are also discussed and developed in the corresponding sections. We note that Theorem 1 (A) and (B) generalize the classical result H sep (B 2 ) = 4 and the result c sep (n, 2) = 2n − 2 √ n , n ≥ 2, proved by Bezdek, Szalkai and Szalkai [7], respectively.…”

Section: Introductionsupporting

confidence: 72%

“…This notion was introduced by G. Fejes Tóth and L. Fejes Tóth [14] and has attracted significant attention. We can define a totally separable packing of translates of a d-dimensional convex body K in a similar way by requiring any two packing elements to be separated by a hyperplane in E d disjoint from the interior of every packing element [6,7]. One can think of a totally separable packing as a packing with barriers.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we introduce the totally separable analogues of the Hadwiger number and the contact number for d-dimensional convex bodies, which we denote by H sep (·) and c sep (·, n, d), respectively, and then mostly study them in the plane. Note that the quantity c sep (n, d) = c sep (B d , n, d) was investigated in [7]. Some results obtained in that paper will be stated and used in the sequel.…”

Section: Introductionmentioning

confidence: 99%

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On contact graphs of totally separable domains

2018

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Contact graphs have emerged as an important tool in the study of translative packings of convex bodies. The contact number of a packing of translates of a convex body is the number of edges in the contact graph of the packing, while the Hadwiger number of a convex body is the maximum vertex degree over all such contact graphs. In this paper, we investigate the Hadwiger and contact numbers of totally separable packings of convex domains, which we refer to as the separable Hadwiger number and the separable contact number, respectively. We show that the separable Hadwiger number of any smooth convex domain is 4 and the maximum separable contact number of any packing of n translates of a smooth strictly convex domain is 2n − 2 √ n . Our proofs employ a characterization of total separability in terms of hemispherical caps on the boundary of a convex body, Auerbach bases of finite dimensional real normed spaces, angle measures in real normed planes, minimal perimeter polyominoes and an approximation of smooth o-symmetric strictly convex domains by, what we call, Auerbach domains.for every x ∈ R d . We denote the corresponding normed space by (R d , · K ). If K = B d , then we denote the norm · B d simply by · . Thus E d = (R d , · ). A d-dimensional convex body K is said to be smooth if at every point on the boundary bd K of K, the body K is supported by a unique hyperplane of E d and strictly convex if the boundary of K contains no nontrivial line segment. Given d-dimensional convex bodies K and L, their Minkowski sum (vector sum) is denoted by K + L and defined aswhich is a convex body in E d .The kissing number problem asks for the maximum number k(d) of non-overlapping translates of B d that can touch B d in E d . Clearly, k(2) = 6. However, the value of k(3) remained a mystery for several hundred years and caused a famous argument between Newton and Gregory in the 17th century. To date, the only *

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Section: Introductionsupporting

confidence: 72%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On contact graphs of totally separable domains

2018

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“…In this paper we investigate the maximum vertex degree (called separable Hadwiger number ), as well as the maximum number of edges (called the maximum separable contact number ) of the contact graphs of totally separable packings by a given number of translates of a smooth or strictly convex body K in E d . This extends and generalizes the results of [4] and [6]. The details follow.…”

Section: Introductionsupporting

confidence: 85%

“…It is mentioned in [6] that based on [9] (see also, [18] and [14]) it follows in a straightforward way that H sep (B d ) = 2d for all d ≥ 2. On the other hand, if K is an o-symmetric convex body in E d , then each facet of the minimum volume circumscribed parallelotope of K touches K at the center of the facet and so, clearly H sep (K) ≥ 2d.…”

Section: Introductionmentioning

confidence: 99%

On contact graphs of totally separable packings in low dimensions

Bezdek

Naszódi

2018

Advances in Applied Mathematics

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The contact graph of a packing of translates of a convex body in Euclidean d-space E d is the simple graph whose vertices are the members of the packing, and whose two vertices are connected by an edge if the two members touch each other. A packing of translates of a convex body is called totally separable, if any two members can be separated by a hyperplane in E d disjoint from the interior of every packing element.We give upper bounds on the maximum vertex degree (called separable Hadwiger number ) and the maximum number of edges (called separable contact number ) of the contact graph of a totally separable packing of n translates of an arbitrary smooth convex body in E d with d = 2, 3, 4. In the proofs, linear algebraic and convexity methods are combined with volumetric and packing density estimates based on the underlying isoperimetric (resp., reverse isoperimetric) inequality.2010 Mathematics Subject Classification. (Primary) 05C10, 52C15, (Secondary) 05B40, 46B20.

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Contact Numbers for Sphere Packings

Bezdek

Khan

2018

Bolyai Society Mathematical Studies

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In discrete geometry, the contact number of a given finite number of non-overlapping spheres was introduced as a generalization of Newton's kissing number. This notion has not only led to interesting mathematics, but has also found applications in the science of self-assembling materials, such as colloidal matter. With geometers, chemists, physicists and materials scientists researching the topic, there is a need to inform on the state of the art of the contact number problem. In this paper, we investigate the problem in general and emphasize important special cases including contact numbers of minimally rigid and totally separable sphere packings. We also discuss the complexity of recognizing contact graphs in a fixed dimension. Moreover, we list some conjectures and open problems.

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On contact numbers of totally separable unit sphere packings

Abstract: Contact numbers are natural extensions of kissing numbers. In this paper we give estimates for the number of contacts in a totally separable packing of n unit balls in Euclidean d-space for all n > 1 and d > 1.

Cited by 14 publications

References 15 publications

On contact graphs of totally separable domains

On contact graphs of totally separable domains

On contact graphs of totally separable packings in low dimensions

Contact Numbers for Sphere Packings

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