Zur Zerlegung von Einheitskugelüberdeckungen des \(E^ d\) in Einheitskugelpackungen. (On the decomposition of coverings by units balls in \(E^ d\) into packings of unit balls) (Q1826121)

Let C be a covering of the d-dimensional Euclidean space (d\(\geq 3)\) by closed unit balls. The covering C is said to be decomposable into k packings if the members of C can be partitioned into k classes, each of which forms a packing in \(E^ d\). Let k(d) be the smallest integer such that there exists a covering C of \(E^ d\) which is decomposable into k(d) packings. It is proved that \(k(3)=k(4)=4\) and \(k(d)\geq d+1,\) for \(d\geq 5\), and it is suggested that \(k(d)>2^{(0.599+o(1))d}\) as \(d\to \infty\). In the case \(d=3\), two different proofs are given.