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The Set of Packing and Covering Densities of Convex Disks

机译:凸盘的包装和覆盖密度的集合

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For every convex disk (K) (a convex compact subset of the plane, with non-void interior), the packing density (delta (K)) and covering density ({vartheta (K)}) form an ordered pair of real numbers, i.e., a point in (mathbb{R }^2). The set (varOmega ) consisting of points assigned this way to all convex disks is the subject of this article. A few known inequalities on (delta (K)) and ({vartheta (K)}) jointly outline a relatively small convex polygon (P) that contains (varOmega ), while the exact shape of (varOmega ) remains a mystery. Here we describe explicitly a leaf-shaped convex region (Lambda ) contained in (varOmega ) and occupying a good portion of (P). The sets (varOmega _T) and (varOmega _L) 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 (varOmega _T) and (varOmega _L) are compact. Furthermore, the sets (varOmega , ,varOmega _T) and (varOmega _L) contain the subsets (varOmega ^star , ,varOmega _T^star ) and (varOmega _L^star ) respectively, corresponding to the centrally symmetric convex disks (K), and our leaf (Lambda ) is contained in each of (varOmega ^star , ,varOmega _T^star ) and (varOmega _L^star ).
机译:对于每个凸圆盘(K)(平面的凸紧致子集,内部无空隙),堆积密度(δ(K))和覆盖密度({vartheta(K)})形成有序的实数对,即(mathbb {R} ^ 2)中的一个点。本文所讨论的主题是由以这种方式分配给所有凸磁盘的点组成的集合(varOmega)。 (delta(K))和({vartheta(K)})上的一些已知不等式共同勾勒出一个相对较小的,包含(varOmega)的凸多边形(P),而(varOmega)的确切形状仍然是个谜。在这里,我们明确描述了包含在(varOmega)中并占据(P)很大一部分的叶形凸区域(Lambda)。类似地定义了平移堆积和覆盖密度以及晶格堆积和覆盖密度的集合(varOmega _T)和(varOmega _L),分别将(K)的允许排列限制为翻译副本或栅格排列。由于平移和晶格密度函数的仿射不变性,集合(varOmega _T)和(varOmega _L)是紧凑的。此外,集合(varOmega,,varOmega _T)和(varOmega _L)分别包含子集(varOmega ^ star,,varOmega _T ^ star)和(varOmega _L ^ star),分别对应于中心对称凸盘(K),并且我们的叶子(Lambda)包含在(varOmega ^ star,,varOmega _T ^ star)和(varOmega _L ^ star)中。

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