The blocking numbers of convex bodies
The blocking number \(b(K)\) of a convex body \(K\) in Euclidean \(n\)-space \(\mathbb{R}^n\) is defined as the smallest possible number of disjoint translates of \(K\) which touch from outside the boundary of \(K\) and which prevent any additional disjoint translate of \(K\) from touching \(K\) from outside. The authors present a lower bound of the blocking number of each centrally symmetric convex body \(C\) based on the so-called \(M\)-curvature of \(C\). They also prove that there are infinitely many dimensions \(n\) such that a centrally symmetric convex body \(C\subset\mathbb{R}^n\) and a subspace \(R\) can be found with \(b(C)\leq b(C\cap R)+1\). It is shown that the blocking number of an \(n\)-dimensional cube is \(2^n\). Moreover, the blocking number of a three-dimensional ball is 6, and of a four-dimensional ball is 9.
scientific article; zbMATH DE number 1507674