Sphere covering by minimal number of caps and short closed sets

TL;DR

This paper establishes the minimal number of short closed sets (caps) needed to cover an n-sphere and characterizes the intersection properties of such coverings, providing a precise combinatorial-geometric result.

Contribution

It proves that exactly n+2 caps are needed to cover an n-sphere and describes the intersection conditions that characterize minimal coverings.

Findings

Minimal number of caps needed is n+2

Intersection of all caps is empty

Any proper subfamily has non-empty intersection

Abstract

A subset of the sphere is said short if it is contained in an open hemisphere. A short closed set which is geodesically convex is called a cap. The following theorem holds: 1. The minimal number of short closed sets covering the $n$-sphere is $n+2$. 2. If $n+2$ short closed sets cover the $n$-sphere then (i) their intersection is empty; (ii) the intersection of any proper subfamily of them is non-empty. In the case of caps (i) and (ii) are also sufficient for the family to be a covering of the sphere.