A Cap Covering Theorem

A cap of spherical radius $\alpha$ on a unit $d$-sphere $S$ is the set points within distance from given point sphere. Let $\mathcal F$ be finite caps lying $S$. We prove that if no hyperplane through center $ S divides into two non-empty subsets without intersecting any in F$, then there equal to sum radii all covering provided less $\pi/2$. This analog so-called Circle Covering Theorem by Goodman and strengthening Fejes T\'oth's zone conjecture proved Jiang author arXiv:1703.10550.