A Cap Covering Theorem

Abstract

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.