论文标题
在平面和高维度中完全可分离的包装中的触点
Contacts in totally separable packings in the plane and in high dimensions
论文作者
论文摘要
我们研究了完全可分离的}包装的接触结构}凸面$ k $ in $ \ mathbb {r}^d $的翻译包装，也就是说，任何两个接触物体都有分离的超平面的包装，这些超平面不会与包装中任何翻译的内部相交。可分离的Hadwiger编号$ h _ {\ text {sep}}（k）$ k $的$被定义为单个翻译触摸的最大翻译数量，最大值的最大分配为$ k $的所有完全可分离的包装。我们证明，对于每$ d \ geq 8 $，都存在一个平滑而严格的凸出$ k $，in $ \ mathbb {r}^d $，带有$ h _ {\ text {sep}}}}（k）> 2d $，and andythictionally，an $ h _ {\ text {sep}}}（k）=ω\ bigl（（3/\ sqrt {8}）^d \ bigr）$。 我们表明，阿隆（Alon）的欧几里得单位球包装，每当$ d $是$ 4 $的功率时，每种翻译都至少触摸$ 2^{\ sqrt {d}} $，可以适应$ \ ell_1 $ - 单位$ -Unit Ball具有相同触摸的属性的完全可分离的包装。 我们还考虑了完全可分开的$ n $翻译包装中的最大接触对数。我们证明，最大值等于$ \ lfloor 2n-2 \ sqrt {n} \ rfloor $，并且仅当$ k $是准六边形，从而完成所有平面凸体的确定此值。
We study the contact structure of totally separable} packings of translates of a convex body $K$ in $\mathbb{R}^d$, that is, packings where any two touching bodies have a separating hyperplane that does not intersect the interior of any translate in the packing. The separable Hadwiger number $H_{\text{sep}}(K)$ of $K$ is defined to be the maximum number of translates touched by a single translate, with the maximum taken over all totally separable packings of translates of $K$. We show that for each $d\geq 8$, there exists a smooth and strictly convex $K$ in $\mathbb{R}^d$ with $H_{\text{sep}}(K)>2d$, and asymptotically, $H_{\text{sep}}(K)=Ω\bigl((3/\sqrt{8})^d\bigr)$. We show that Alon's packing of Euclidean unit balls such that each translate touches at least $2^{\sqrt{d}}$ others whenever $d$ is a power of $4$, can be adapted to give a totally separable packing of translates of the $\ell_1$-unit ball with the same touching property. We also consider the maximum number of touching pairs in a totally separable packing of $n$ translates of any planar convex body $K$. We prove that the maximum equals $\lfloor 2n-2\sqrt{n}\rfloor$ if and only if $K$ is a quasi hexagon, thus completing the determination of this value for all planar convex bodies.
