Sphere covering by minimal number of caps and short closed sets
A. B. N\'emeth

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 -sphere is . 2. If short closed sets cover the -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.
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Taxonomy
TopicsAdvanced Numerical Analysis Techniques · Computational Geometry and Mesh Generation
