从可分离Tammes问题到单位球面上大圆的极值分布
From the separable Tammes problem to extremal distributions of great circles in the unit sphere
AI总结：
本文研究单位球面上全等球冠的完全可分离堆积与覆盖问题，解决了最多8个球冠的可分离Tammes问题，证明三维欧氏球的中心可分离亲吻数为8，并给出大圆铺砌胞腔最小内切半径及TS-覆盖的界，部分结果推广到高维球面。
摘要：
二维单位球面 $\mathbb{S}^2$ 上的一族球冠被称为完全可分离堆积（简称TS-堆积），如果任意两个球冠都能被一个大圆分离，且该大圆与堆积中每个球冠的内部都不相交。可分离Tammes问题要求确定在 $\mathbb{S}^2$ 中由给定数量的全等球冠构成TS-堆积的最大密度。我们解决了最多 $8$ 个球冠的情形，并根据球冠的角半径给出了全等球冠TS-堆积密度的上界。基于此，我们证明了 $3$ 维欧氏球的中心可分离亲吻数为 $8$。此外，我们证明了由 $\mathbb{S}^2$ 中 $n>1$ 个大圆生成的铺砌的胞腔最小内切半径的最大值的界。接下来，我们证明了 $\mathbb{S}^2$ 上由全等球冠构成的TS-覆盖的对偶界。这里，$\mathbb{S}^2$ 上的球冠覆盖被称为完全可分离覆盖（简称TS-覆盖），如果存在由 $\mathbb{S}^2$ 上有限多个大圆生成的铺砌，使得该铺砌的胞腔被覆盖中两两不同的球冠所覆盖。最后，我们将关于TS-覆盖的一些界推广到维数 $>2$ 的球面空间。
英文摘要：
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 $>2$.
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