On the density of finite packings
Let \({\mathcal C}^ d\) denote the set of compact convex sets in the Euclidean d-space \(E^ d\). Let V(K) be the volume of any \(K\in {\mathcal C}^ d\). Further let \({\mathcal C}^ d_ k\) denote the set of \(K\in {\mathcal C}^ d\) which contains k translates \(B^ d_ 1,...,B^ d_ k\) of \(B^ d\) with \(B^ d_ i\cap B^ d_ j=\emptyset\) for \(i\neq j\). Here \(B^ d\) is the unit ball and \(V(B^ d)=\omega_ d\). For each \(K\in {\mathcal C}^ d_ k\) let \(\delta (K)=k\omega_ d/V(K),\) and put \(\delta^ d_ k=\max_{K\in {\mathcal C}^ d_ k}\delta (K)=\max_{K\in {\mathcal C}^ d_ k}\{k\omega_ d/V(K)\}.\) The following result is proved: There exist P, \(Q\in {\mathcal C}^ 3\) such that \(P+B^ 3\in {\mathcal C}^ 3_{55},\) \(Q+B_ 3\in {\mathcal C}^ 3_{56}\) with \(\delta (P+B^ 3)=0.6699<0.6707=\delta (S_{55}+B^ 3)\leq S^ 3_{55}\) and \(\delta (S_{56}+B^ 3)=0.6707<0.6710=\delta (Q+B^ 3)\leq \delta^ 3_{56}\) if the centres of the translates lie on a line segment \(S_ k.\) \(S_ k+B^ d\) forms a ''sausage'' with \(V(S_ k+B^ d)=2(k- 1)\omega_{d-1}+\omega_ d.\)
scientific article; zbMATH DE number 3942359