A cap covering theorem

A cap of spherical radius $\alpha$ on a unit $2$-sphere $S$ is the set of points within spherical distance $\alpha$ from a given point on the sphere. Let $\mathcal S$ be a finite set of caps lying on $S$. We prove that if there is no great circle non-intersecting caps of $\mathcal S$ and dividing $\mathcal S$ into two non-empty subsets, then there is a cap of radius equal to the total radius of caps of $\mathcal S$ covering all caps of $\mathcal S$ provided that the total radius is less $\pi/2$. This is the spherical analog of the so-called Circle Covering Theorem by Goodman and Goodman and the strengthening of Fejes T\'oth's zone conjecture proved by Jiang and the author arXiv:1703.10550.Comment: 5 pages, 2 figures. Key words: Bang's plank theore