On the Multiple Covering Densities of Triangles (original) (raw)
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On the Multiple Packing Densities of Triangles
Discrete & Computational Geometry, 2015
Given a convex disk K and a positive integer k, let δ k T (K) and δ k L (K) denote the k-fold translative packing density and the k-fold lattice packing density of K, respectively. Let T be a triangle. In a very recent paper [2], I proved that δ k L (T) = 2k 2 2k+1. In this paper, I will show that δ k T (T) = δ k L (T).
On the Multiple Packing Densities of Triangles Kirati Sriamorn
2016
Given a convex disk K and a positive integer k, let δ T (K) and δ k L(K) denote the k-fold translative packing density and the k-fold lattice packing density of K, respectively. Let T be a triangle. In a very recent paper [2], I proved that δ L(T ) = 2k 2 2k+1 . In this paper, I will show that δ T (T ) = δ k L(T ).
On the Lattice Packings and Coverings of Convex
2014
It is well known that the lattice packing density and the lattice covering density of a triangle are 2 3 and 3 2 respectively [3]. We also know that the lattices that attain these densities both are unique. Let δL(K) and ϑL(K) denote the lattice packing density and the lattice covering density of K, respectively. In this paper, I study the lattice packings and coverings for a special class of convex disks, which includes all triangles and convex quadrilaterals. In particular, I determine the densities δL(Q) and ϑL(Q), where Q is an arbitrary convex quadrilateral. Furthermore, I also obtain all of lattices that attain these densities. Finally, I show that δL(Q)ϑL(Q) ≥ 1 and 1 δ L (Q) + 1 ϑ L (Q) ≥ 2, for each convex quadrilateral Q.
On the Covering Densities of Quarter-Convex Disks
Discrete & Computational Geometry, 2015
It is conjectured that for every convex disks K, the translative covering density of K and the lattice covering density of K are identical. It is well known that this conjecture is true for every centrally symmetric convex disks. For the non-symmetric case, we only know that the conjecture is true for triangles [1]. In this paper, we prove the conjecture for a class of convex disks (quarter-convex disks), which includes all triangles and convex quadrilaterals.
On the Lattice Packings and Coverings of the Plane with Convex Quadrilaterals
arXiv (Cornell University), 2014
It is well known that the lattice packing density and the lattice covering density of a triangle are 2 3 and 3 2 respectively [3]. We also know that the lattices that attain these densities both are unique. Let δL(K) and ϑL(K) denote the lattice packing density and the lattice covering density of K, respectively. In this paper, I study the lattice packings and coverings for a special class of convex disks, which includes all triangles and convex quadrilaterals. In particular, I determine the densities δL(Q) and ϑL(Q), where Q is an arbitrary convex quadrilateral. Furthermore, I also obtain all of lattices that attain these densities. Finally, I show that δL(Q)ϑL(Q) ≥ 1 and 1 δ L (Q) + 1 ϑ L (Q) ≥ 2, for each convex quadrilateral Q.
Packing and Covering with Centrally Symmetric Convex Disks
Discrete & Computational Geometry, 2014
Given a convex disk K (a convex compact planar set with nonempty interior), let δ L (K) and θ L (K) denote the lattice packing density and the lattice covering density of K, respectively. We prove that for every centrally-symmetric convex disk K we have that 1 ≤ δ L (K)θ L (K) ≤ 1.17225. .. The left inequality is tight and it improves a 10-year old result. Keywords Arrangements of convex disks • Packing density • Covering density 1 Introduction In this paper, we consider arrangements of convex disks in the Euclidean plane. A convex disk is a compact convex set with nonempty interior; its area will be denoted by A(K). An arrangement of congruent copies (translates) of a convex disk K is a family A of convex disks, each of which is congruent to (is a translate of) K. The arrangement is a packing if its members' interiors are mutually disjoint, and it is a covering if the union of its members is the whole plane.
Covering the plane with copies of a convex disk
Discrete & Computational Geometry, 1998
We prove that for every convex disk C in the plane there exists a double-lattice covering of the plane with copies of C with density 0 < 1.2281772. This improves the best previously known upper bound 0 < 8/(3 + 24'-3) ~ 1.2376043, due to Kuperberg, but it is still far from the conjectured value t9 = 27r/3~,'3 ~ 1.2091993.
The Set of Packing and Covering Densities of Convex Disks
Discrete & Computational Geometry, 2013
For every convex disk K (a convex compact subset of the plane, with non-void interior), the packing density δ(K) and covering density ϑ(K) form an ordered pair of real numbers, i.e., a point in R 2. The set Ω consisting of points assigned this way to all convex disks is the subject of this article. A few known inequalities on δ(K) and ϑ(K) jointly outline a relatively small convex polygon P that contains Ω, while the exact shape of Ω remains a mystery. Here we describe explicitly a leaf-shaped convex region Λ contained in Ω and occupying a good portion of P. The sets ΩT and ΩL of translational packing and covering densities and lattice packing and covering densities are defined similarly, restricting the allowed arrangements of K to translated copies or lattice arrangements, respectively. Due to affine invariance of the translative and lattice density functions, the sets ΩT and ΩL are compact. Furthermore, the sets Ω, ΩT and ΩL contain the subsets Ω , Ω T and Ω L respectively, corresponding to the centrally symmetric convex disks K,
Discrete & Computational Geometry, 2001
Givenafamily C of planeconvexbodies,let n (C) bethesetof all pairs(x, y) with the propertythatthereexistsK E C suchthat iJ(K) = x and8(K) = y, whereiJ(K) and8(K) denotethedensitiesofthethinnestcoveringandthedensest packingoftheplane with copiesof K, respectively.The set nL(C) is definedanalogously, with the difference thatwe restrictour attentionto lattice packingsandcoverings. We provethat,for everycentrallysymmetricplaneconvexbody K, and give an exactanalytic descriptionof ndP s) whereP s is the family of all centrally symmetricoctagons. This allows us to showthatthe aboveinequalitiesareasymptotically tight.