On \(k^+\)-neighbour packings and one-sided Hadwiger configurations (Q1403517)

Let \(K\) be a convex body with nonempty interior in Euclidean \(d\)-space. The authors show that the maximum number of non-overlapping translates of \(K\) that can touch \(K\) and can lie in a closed supporting half-space of that body is at most \(2\cdot 3^{d-1}-1\), with equality only for \(K\) an affine image of a \(d\)-cube. Such configurations are interesting since they occur for example in the ``boundary regions'' of finite packings. A packing is called a \(k^+\)-neighbour packing if any packing element has at least \(k\) neighbours. From the result above it follows then that in Euclidean \(d\)-space any \(k^+\)-neighbour packing by translates of a non-degenerate convex body has positive density for all \(k\) not smaller than \(2\cdot 3^{d-1}\), and that there is a \((2\cdot 3^{d-1})^+\)-neighbour packing by translates of a \(d\)-cube having density 0.