Density bounds for outer parallel domains of unit ball packings

The Kneser-Poulsen Conjecture states that if the centers of a family of N unit balls in E d is contracted, then the volume of the union (resp., intersection) does not increase (resp., decrease). We consider two types of special contractions.First, a uniform contraction is a contraction where all the pairwise distances in the first set of centers are larger than all the pairwise distances in the second set of centers. We obtain that a uniform contraction of the centers does not decrease the volume of the intersection of the balls, provided that N ≥ (1 + √ 2) d . Our result extends to intrinsic volumes. We prove a similar result concerning the volume of the union.Second, a strong contraction is a contraction in each coordinate. We show that the conjecture holds for strong contractions. In fact, the result extends to arbitrary unconditional bodies in the place of balls.2010 Mathematics Subject Classification. 52A20,52A22.

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“…3.1. To prove part (a) in Theorem 3.1, we note that Theorem 2 of [BL15] implies in a straightforward way that…”

Section: Proof Of Theorem 12mentioning

confidence: 99%

“…For the proof of part (b), we use a theorem of Rogers, discussed in the introduction of [BL15], according to which…”

Section: 2mentioning

confidence: 99%

The Kneser–Poulsen Conjecture for Special Contractions

Bezdek

Naszódi

2018

Discrete Comput Geom

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“…Interestingly enough one can interpret the contact number problem on the exact values of c(n, d) as a volume minimization question. Here we give only an outline of that idea introduced and discussed in detail in [14].…”

Section: Packings By Translates Of a Convex Bodymentioning

confidence: 99%

“…On the other hand, it was proved in [9] (see also Corollary 7.4) that δ d = δ d (λ) for all λ ≥ 1 leading to the classical sphere packing problem. Now, we are ready to put forward the following question from [14].…”

Section: Packings By Translates Of a Convex Bodymentioning

confidence: 99%

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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“…We note that in [4] the covering ratio just introduced was called soft density. We prefer to use the term covering ratio in order to emphasize that it means the fraction of plane covered by the soft elements of the given soft packing.…”

Section: On the Smallest Area Convex Hull Of Totally Separable Translmentioning

confidence: 99%

Bounds for Totally Separable Translative Packings in the Plane

Bezdek

Lángi

2018

Discrete Comput Geom

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A packing of translates of a convex domain in the Euclidean plane is said to be totally separable if any two packing elements can be separated by a line disjoint from the interior of every packing element. This notion was introduced by G. Fejes Tóth and L. Fejes Tóth (1973) and has attracted significant attention. In this paper we prove an analogue of Oler's inequality for totally separable translative packings of convex domains and then we derive from it some new results. This includes finding the largest density of totally separable translative packings of an arbitrary convex domain and finding the smallest area convex hull of totally separable packings (resp., totally separable soft packings) generated by given number of translates of a convex domain (resp., soft convex domain). Finally, we determine the largest covering ratio (that is, the largest fraction of the plane covered by the soft disks) of an arbitrary totally separable soft disk packing with given soft parameter.

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Density bounds for outer parallel domains of unit ball packings

Abstract: We give upper bounds for the density of unit ball packings relative to their outer parallel domains and discuss their connection to contact numbers. Also, packings of soft balls are introduced and upper bounds are given for the fraction of space covered by them.Comment: 22 pages, 1 figur

Cited by 9 publications

References 29 publications

The Kneser–Poulsen Conjecture for Special Contractions

The Kneser–Poulsen Conjecture for Special Contractions

Contact Numbers for Sphere Packings

Bounds for Totally Separable Translative Packings in the Plane

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