论文标题
从可分离的毛茸茸问题到单位球中大圆圈的极端分布
From the separable Tammes problem to extremal distributions of great circles in the unit sphere
论文作者
论文摘要
二维单元球的一个球形帽子$ \ mathbb {s}^2 $很简单地称为完全可分开的包装，如果任何两个球形盖可以通过一个大圆圈隔开，这与包装中每个球形盖的内部脱节。可分离的喃喃自语问题要求给定数量的一致球形盖，形成$ \ mathbb {s}^2 $中的TS包装。我们解决了该问题的最高$ 8 $球形盖，并且在其角度半径上，一致球形盖的任何TS包装的密度。基于此，我们证明了$ 3 $维的欧几里得球的中心可分开的接吻数为$ 8 $。此外，我们证明了界限的最大范围，即$ n> 1 $ great Circles在$ \ mathbb {s}^2 $中生成的砖块的最小inradius。接下来，我们证明了$ \ mathbb {s}^2 $的TS覆盖的双重界限。在这里，由球形帽覆盖$ \ mathbb {s}^2 $，称为完全可分开的覆盖物，简而言之，如果存在由$ \ mathbb {s}^2 $有限的大圆圈产生的瓷砖，则覆盖范围，覆盖有限的圆圈，因此，覆盖层的覆盖层的细胞覆盖了覆盖的覆盖范围的覆盖范围。最后，我们将TS覆盖的一些界限扩展到尺寸$> 2 $的球形空间。
A family of spherical caps of the 2-dimensional unit sphere $\mathbb{S}^2$ is called a totally separable packing in short, a TS-packing if any two spherical caps can be separated by a great circle which is disjoint from the interior of each spherical cap in the packing. The separable Tammes problem asks for the largest density of given number of congruent spherical caps forming a TS-packing in $\mathbb{S}^2$. We solve this problem up to $8$ spherical caps and upper bound the density of any TS-packing of congruent spherical caps in terms of their angular radius. Based on this, we show that the centered separable kissing number of $3$-dimensional Euclidean balls is $8$. Furthermore, we prove bounds for the maximum of the smallest inradius of the cells of the tilings generated by $n>1$ great circles in $\mathbb{S}^2$. Next, we prove dual bounds for TS-coverings of $\mathbb{S}^2$ by congruent spherical caps. Here a covering of $\mathbb{S}^2$ by spherical caps is called a totally separable covering in short, a TS-covering if there exists a tiling generated by finitely many great circles of $\mathbb{S}^2$ such that the cells of the tiling are covered by pairwise distinct spherical caps of the covering. Finally, we extend some of our bounds on TS-coverings to spherical spaces of dimension $>2$.
