A cap covering theorem

A cap of spherical radius $\alpha$ on a unit $d$-sphere $S$ is the set of points within spherical distance $\alpha$ 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 $\pi/2$. 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.