一个帽覆盖定理
A cap covering theorem
摘要：
单位$d$上的球面半径为$α$的帽-球面$S$是从球面上的给定点到球面距离$α$内的点的集合。设$\mathcal F$是位于$S$上的帽的有限集。证明了如果没有通过$ S $中心的超平面将$\mathcal F$划分为两个非空子集，且不与$\mathcal F$中的任何帽相交，则存在一个半径等于$\mathcal F$中所有帽的半径之和的帽覆盖$\mathcal F$的所有帽，只要半径之和小于$π/2$ .这是Goodman和Goodman的所谓圆覆盖定理的球面模拟，以及Jiang和作者arXiv：1703.10550证明的Fejes Tóth区域猜想的加强。
英文摘要：
A cap of spherical radius $α$ on a unit $d$-sphere $S$ is the set of points within spherical distance $α$ from a given point on the sphere. Let $\mathcal F$ be a finite set of caps lying on $S$. We prove that if no hyperplane through the center of $ S $ divides $\mathcal F$ into two non-empty subsets without intersecting any cap in $\mathcal F$, then there is a cap of radius equal to the sum of radii of all caps in $\mathcal F$ covering all caps of $\mathcal F$ provided that the sum of radii is less $π/2$. This is the spherical analog of the so-called Circle Covering Theorem by Goodman and Goodman and the strengthening of Fejes Tóth's zone conjecture proved by Jiang and the author arXiv:1703.10550.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
