On the Multiple Covering Densities of Triangles (original) (raw)
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.
On the covering index of convex bodies
Aequationes mathematicae, 2016
Covering a convex body by its homothets is a classical notion in discrete geometry that has resulted in a number of interesting and long-standing problems. Swanepoel introduced the covering parameter of a convex body as a means of quantifying its covering properties. In this paper, we introduce two relatives of the covering parameter called covering index and weak covering index, which upper bound well-studied quantities like the illumination number, the illumination parameter and the covering parameter of a convex body. Intuitively, the two indices measure how well a convex body can be covered by a relatively small number of homothets having the same relatively small homothety ratio. We show that the covering index is a lower semicontinuous functional on the Banach-Mazur space of convex bodies. We further show that the affine d-cubes minimize covering index in any dimension d, while circular disks maximize it in the plane. Furthermore, the covering index satisfies a nice compatibility with the operations of direct vector sum and vector sum. In fact, we obtain an exact formula for the covering index of a direct vector sum of convex bodies that works in infinitely many instances. This together with a minimization property can be used to determine the covering index of infinitely many convex bodies. As the name suggests, the weak covering index loses some of the important properties of the covering index. Finally, we obtain upper bounds on the covering and weak covering index.
New circle coverings of an equilateral triangle
1997
Recently, Melissen has determined the thinnest coverings of an equilateral triangle with 1,...,6, and 10 equal circles; see [5,6]. Here, we will determine the thinnest covering with nine circles, and give a new, short prooffor the thinnest covering with ten circles. Furthermore, we provide thin coverings with up to eighteen circles for the remaining cases. These coverings are conjectured to be optimal.
On convex segments in a triangulation
Discrete Mathematics, 1993
A segment (= l-cell) of a planar triangulation 0 is conuex if it is common to two triangles (2-cells) whose union is a convex set. We determine the maximal number of convex segments of a triangulation over all triangulations CJ having n boundary vertices and m inner vertices (n > 3, m >O).
An isoperimetric inequality for planar triangulations
2016
We prove a discrete analogue to a classical isoperimetric theorem of Weil for surfaces with non-positive curvature. It is shown that hexagons in the triangular lattice have maximal volume among all sets of a given boundary in any triangulation with minimal degree 6.
On well-covered triangulations: Part I
Discrete Applied Mathematics, 2003
A graph G is said to be well-covered if every maximal independent set of vertices has the same cardinality. A planar (simple) graph in which each face is a triangle is called a triangulation. It is the aim of this paper to prove that there are no 5-connected planar well-covered triangulations. ?