Bounds for Totally Separable Translative Packings in the Plane

Bezdek, Károly; Lángi, Zsolt

doi:10.1007/s00454-018-0029-6

Cited by 4 publications

(5 citation statements)

References 13 publications

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“…Finally, we conclude our investigations with some higher dimensional analogue statements. We note that totally separable packings and coverings have been already studied in Euclidean (resp., normed) spaces ( [2], [3], [4], [5], [6], [7], [8], [9], [16], [17], [26]), but not yet in spherical spaces, which is the main target of this paper. The details are as follows.…”

Section: Introductionmentioning

confidence: 99%

From the separable Tammes problem to extremal distributions of great circles in the unit sphere

Bezdek¹,

Lángi²

2022

Preprint

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A family of spherical caps of the 2-dimensional unit sphere S 2 is called a totally separable packing in short, a TS-packing if any two spherical caps can be separated by a great circle which is disjoint from the interior of each spherical cap in the packing. The separable Tammes problem asks for the largest density of given number of congruent spherical caps forming a TS-packing in S 2 . We solve this problem up to 8 spherical caps and upper bound the density of any TS-packing of congruent spherical caps in terms of their angular radius. Furthermore, we prove bounds for the maximum of the smallest inradius of the cells of the tilings generated by n > 1 great circles in S 2 . Next, we prove dual bounds for TS-coverings of S 2 by congruent spherical caps. Here a covering of S 2 by spherical caps is called a totally separable covering in short, a TS-covering if there exists a tiling generated by finitely many great circles of S 2 such that the cells of the tiling are covered by pairwise distinct spherical caps of the covering. Finally, we extend some of our bounds on TS-coverings to spherical spaces of dimension > 2.

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Section: Introductionmentioning

confidence: 99%

From the separable Tammes problem to extremal distributions of great circles in the unit sphere

Bezdek¹,

Lángi²

2022

Preprint

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“…The problem investigated in this paper is similar in nature to those dealing with the volume of the convex hull of a family of convex bodies, which has a rich literature. This includes a result of Oler [17] (see also [3]), which is also of lattice geometric origin [21], and the notion of parametric density of Betke, Henk and Wills [1]. In particular, our problem is closely related to the notion of density with respect to outer parallel domains defined in [3].…”

Section: Introductionmentioning

confidence: 99%

“…This includes a result of Oler [17] (see also [3]), which is also of lattice geometric origin [21], and the notion of parametric density of Betke, Henk and Wills [1]. In particular, our problem is closely related to the notion of density with respect to outer parallel domains defined in [3]. Applications of A c c e p t e d m a n u s c r i p t (generalized) Minkowski arrangements in other branches of mathematics can be found in [19,20].…”

Section: Introductionmentioning

confidence: 99%

On generalized Minkowski arrangements

Kadlicskó

Lángi

2022

Ars Math. Contemp.

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The concept of a Minkowski arrangement was introduced by Fejes Tóth in 1965 as a family of centrally symmetric convex bodies with the property that no member of the family contains the center of any other member in its interior. This notion was generalized by Fejes Tóth in 1967, who called a family of centrally symmetric convex bodies a generalized Minkowski arrangement of order µ for some 0 < µ < 1 if no member K of the family overlaps the homothetic copy of any other member K ′ with ratio µ and with the same center as K ′ . In this note we prove a sharp upper bound on the total area of the elements of a generalized Minkowski arrangement of order µ of finitely many circular disks in the Euclidean plane. This result is a common generalization of a similar result of Fejes Tóth for Minkowski arrangements of circular disks, and a result of Böröczky and Szabó about the maximum density of a generalized Minkowski arrangement of circular disks in the plane. In addition, we give a sharp upper bound on the density of a generalized Minkowski arrangement of homothetic copies of a centrally symmetric convex body.

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“…The problem investigated in this paper is similar in nature to those dealing with the volume of the convex hull of a family of convex bodies, which has a rich literature. This includes a result of Oler [16] (see also [3]), which is also of lattice geometric origin [20], and the notion of parametric density of Betke, Henk and Wills [1]. In particular, our problem is closely related to the notion of density with respect to outer parallel domains defined in [3].…”

Section: Introductionmentioning

confidence: 99%

“…This includes a result of Oler [16] (see also [3]), which is also of lattice geometric origin [20], and the notion of parametric density of Betke, Henk and Wills [1]. In particular, our problem is closely related to the notion of density with respect to outer parallel domains defined in [3]. Applications of (generalized) Minkowski arrangements in other branches of mathematics can be found in [18,19].…”

Section: Introductionmentioning

confidence: 99%

On generalized Minkowski arrangements

Kadlicskó¹,

Lángi²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

The concept of a Minkowski arrangement was introduced by Fejes Tóth in 1965 as a family of centrally symmetric convex bodies with the property that no member of the family contains the center of any other member in its interior. This notion was generalized by Fejes Tóth in 1967, who called a family of centrally symmetric convex bodies a generalized Minkowski arrangement of order µ for some 0 < µ < 1 if no member K of the family overlaps the homothetic copy of any other member K with ratio µ and concentric with K . In this note we prove a sharp upper bound on the total area of the elements of a generalized Minkowski arrangement of order µ of finitely many circular disks in the Euclidean plane. This result is a common generalization of a similar result of Fejes Tóth for Minkowski arrangements of circular disks, and a result of Böröczky and Szabó about the maximum density of a generalized Minkowski arrangement of circular disks in the plane. In addition, we give a sharp upper bound on the density of a generalized Minkowski arrangement of homothetic copies of a centrally symmetric convex body.

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Bounds for Totally Separable Translative Packings in the Plane

Cited by 4 publications

References 13 publications

From the separable Tammes problem to extremal distributions of great circles in the unit sphere

From the separable Tammes problem to extremal distributions of great circles in the unit sphere

On generalized Minkowski arrangements

On generalized Minkowski arrangements

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