Finite and uniform stability of sphere coverings (Q1892412)

A ball covering of Euclidean \(d\)-space \(E^d\) is called \(n\)-stable if no subset of \(n\) balls can be moved such that the covering property is maintained. The covering is said to be finitely stable if it is \(n\)- stable for every positive integer \(n\). The authors prove that the thinnest cubic-lattice ball covering of \(E^d\) is not finitely stable. The proof is provided using the so-called cabling method which deals with a number of cable frameworks connecting centers of the balls and some intersections of the balls. It is shown that a sphere covering of \(E^d\) with unit balls whose centers form a discrete point set is finitely stable if and only if the corresponding cable frameworks are finitely rigid.