573 research outputs found
On the Multiple Packing Densities of Triangles
Given a convex disk and a positive integer , let and
denote the -fold translative packing density and the
-fold lattice packing density of , respectively. Let be a triangle.
In a very recent paper, K. Sriamorn proved that
. In this paper, I will show that
.Comment: arXiv admin note: text overlap with arXiv:1412.539
On the Multiple Covering Densities of Triangles
Given a convex disk and a positive integer , let
and denote the -fold translative covering density and the
-fold lattice covering density of , respectively. Let be a triangle.
In a very recent paper, K. Sriamorn proved that
. In this paper, we will show that
On the Covering Densities of Quarter-Convex Disks
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. In this paper, we prove the conjecture for a class of convex disks
(quarter-convex disks), which includes all triangles and convex quadrilaterals
Multiple coverings with closed polygons
A planar set is said to be cover-decomposable if there is a constant
such that every -fold covering of the plane with translates of
can be decomposed into two coverings. It is known that open convex polygons are
cover-decomposable. Here we show that closed, centrally symmetric convex
polygons are also cover-decomposable. We also show that an infinite-fold
covering of the plane with translates of can be decomposed into two
infinite-fold coverings. Both results hold for coverings of any subset of the
plane.Comment: arXiv admin note: text overlap with arXiv:1009.4641 by other author
Making Octants Colorful and Related Covering Decomposition Problems
We give new positive results on the long-standing open problem of geometric
covering decomposition for homothetic polygons. In particular, we prove that
for any positive integer k, every finite set of points in R^3 can be colored
with k colors so that every translate of the negative octant containing at
least k^6 points contains at least one of each color. The best previously known
bound was doubly exponential in k. This yields, among other corollaries, the
first polynomial bound for the decomposability of multiple coverings by
homothetic triangles. We also investigate related decomposition problems
involving intervals appearing on a line. We prove that no algorithm can
dynamically maintain a decomposition of a multiple covering by intervals under
insertion of new intervals, even in a semi-online model, in which some coloring
decisions can be delayed. This implies that a wide range of sweeping plane
algorithms cannot guarantee any bound even for special cases of the octant
problem.Comment: version after revision process; minor changes in the expositio
Making Triangles Colorful
We prove that for any point set P in the plane, a triangle T, and a positive
integer k, there exists a coloring of P with k colors such that any homothetic
copy of T containing at least ck^8 points of P, for some constant c, contains
at least one of each color. This is the first polynomial bound for range spaces
induced by homothetic polygons. The only previously known bound for this
problem applies to the more general case of octants in R^3, but is doubly
exponential.Comment: 6 page
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