10 research outputs found

### On the Multiple Packing Densities of Triangles

Given a convex disk $K$ and a positive integer $k$, let $\delta_T^k(K)$ and
$\delta_L^k(K)$ 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, K. Sriamorn proved that
$\delta_L^k(T)=\frac{2k^2}{2k+1}$. In this paper, I will show that
$\delta_T^k(T)=\delta_L^k(T)$.Comment: arXiv admin note: text overlap with arXiv:1412.539

### 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

### On the multiple illumination numbers of convex bodies

In this paper, we introduce an $m$-fold illumination number $I^m(K)$ of a
convex body $K$ in Euclidean space $\mathbb{E}^d$, which is the smallest number
of directions required to $m$-fold illuminate $K$, i.e., each point on the
boundary of $K$ is illuminated by at least $m$ directions. We get a lower bound
of $I^m(K)$ for any $d$-dimensional convex body $K$, and get an upper bound of
$I^m(\mathbb{B}^d)$, where $\mathbb{B}^d$ is a $d$-dimensional unit ball. We
also prove that $I^m(K)=2m+1$, for a $2$-dimensional smooth convex body $K$.
Furthermore, we obtain some results related to the $m$-fold illumination
numbers of convex polygons and cap bodies of $\mathbb{B}^d$ in small
dimensions. In particular, we show that $I^m(P)=\left\lceil
mn/{\left\lfloor\frac{n-1}{2}\right\rfloor}\right\rceil$, for a regular convex
$n$-sided polygon $P$

### On the Multiple Covering Densities of Triangles

Given a convex disk $K$ and a positive integer $k$, let $\vartheta_T^k(K)$
and $\vartheta_L^k(K)$ denote the $k$-fold translative covering density and the
$k$-fold lattice covering density of $K$, respectively. Let $T$ be a triangle.
In a very recent paper, K. Sriamorn proved that
$\vartheta_L^k(T)=\frac{2k+1}{2}$. In this paper, we will show that
$\vartheta_T^k(T)=\vartheta_L^k(T)$

### Characterization of the Three-Dimensional Fivefold Translative Tiles

This paper proves the following statement: If a convex body can form a
fivefold translative tiling in $\mathbb{E}^3$, it must be a parallelotope, a
hexagonal prism, a rhombic dodecahedron, an elongated dodecahedron, a truncated
octahedron, a cylinder over a particular octagon, or a cylinder over a
particular decagon, where the octagon and the decagon are fivefold translative
tiles in $\mathbb{E}^2$. Furthermore, it presents an example of multiple tiles
in $\mathbb{E}^3$ with multiplicity at most 10 which is neither a
parallelohedron nor a cylinder