An example concerning the translative kissing number of a convex body (Q1334931)

The translative kissing number \(N_ T(K)\) of the convex body \(K\) is the largest \(N\) such that \(N\) translates of \(K\) with disjoint interiors can all touch \(K\). The lattice kissing number \(N_ L(K)\) is the largest such that \(N\) translates of \(K\) in a lattice packing can all touch \(K\). In 1961, Grünbaum showed that \(N_ T(K)= N_ L(K)\) for all \(K\) in \(E^ 2\); in 1971 Watson found a counterexample for the sphere in \(E^ 9\). The author presents a nice counterexample for a convex polyhedron in \(E^ 3\). The unit cube touches 26 translates in the face-to-face lattice packing, eight of them by a tenuous vertex-to-vertex contact. By slightly faceting four vertices (the endpoints of two opposite edges), the author creates a polyhedron that cannot have more than 22 neighbours in a lattice packing. However, a non-lattice configuration is given in which the polyhedron touches 24 translates.