A cap covering theorem

TL;DR

This paper proves a spherical covering theorem for caps on a sphere, extending classical circle covering results and providing conditions under which a single large cap covers a finite set of smaller caps.

Contribution

It introduces a spherical analog of the Circle Covering Theorem, strengthening previous zone conjecture results and offering new geometric covering conditions.

Findings

A cap of radius equal to the sum of smaller caps covers all if the sum is less than π/2.

The theorem generalizes classical circle covering results to higher-dimensional spheres.

Provides a new geometric criterion for covering caps on a sphere.

Abstract

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.