A cap covering theorem (Q2064753)

A collection of spherical caps on \(\mathbb{S}^d\) is called non-separable if there do not exist hyperplanes which divide this collection into two non-empty sets. In the paper under review, the author proves that if the caps of the collection have radiuses \(\alpha_1,\ldots,\alpha_n\) and \(\alpha_1+\cdots+\alpha_n <\pi/2\), then the whole collection can be covered by a single cap of radius \(\alpha_1+\cdots+\alpha_n\). This result is a spherical analog of the so-called circle covering theorem by \textit{A. W. Goodman} and \textit{R. E. Goodman} [Am. Math. Mon. 52, 494--498 (1945; Zbl 0060.34902)] (conjectured before by Erdős).