Properties of \(\mathcal R\)-sausages (Q1889916)

For a finite set \(V\) in Euclidean space \({\mathbb R}^d\), the Steiner ratio \(\rho(V)\) is the (Euclidean) length of a Steiner minimal tree of \(V\) divided by the length of a minimum spanning tree of \(V\). The paper is concerned with \(\rho_d\), the infimum of \(\rho(V)\) over all finite subsets of \({\mathbb R}^d\). It is known that \(\rho_2=\sqrt{3}/2\), and it has been conjectured that for \(d>2\) (at least if \(d\) is not too large), \(\rho_d\) is approached by the values of \(\rho\) on increasing finite subsets of a certain countable point set, the `\(d\)-sausage'. This consists of the centres of the balls obtained as follows. Start with a unit ball, and successively add unit balls so that the \(N\)th ball touches the \(\min\{d,N-1\}\) most recently added balls. The authors make an extensive, partially computational, study of these objects in dimensions two (`flat sausage') and three (`\({\mathcal R}\)-sausage'), of the Steiner minimal trees of their \(N\)th sections, and of the behaviour of the latter for \(N\to\infty\).