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 \({\cal F}\) be a finite set of caps lying on S. We prove that if no hyperplane through the center of S divides \({\cal F}\) into two non-empty subsets without intersecting any cap in \({\cal F}\), then there is a cap of radius equal to the sum of radii of all caps in \({\cal F}\) covering all caps of \({\cal F}\) provided that the sum of radii is less than 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.

甲帽球面半径的α上的单元d -sphere小号是设定球面距离内的点的α从球体上的给定点。设\({\cal F}\)是位于S上的有限大写集合。我们证明，如果没有通过S中心的超平面将\({\cal F}\)分成两个非空子集而不与\({\cal F}\) 中的任何上限相交，那么存在一个半径相等的上限到全部大写的半径的总和中的\（{\ CAL F} \）覆盖的全部大写\（{\ CAL F} \）提供的半径之和小于π/ 2。

这是古德曼和古德曼所谓的圆覆盖定理的球面模拟，也是蒋和作者证明的费耶斯·托特区域猜想的加强。