The Steiner ratio for the dual normed plane (Q1363707)

The Steiner ratio \(\rho\) of the real plane \(\mathbb{R}^2\) is the infimum of \(L_S(X)\cdot L_M(X)^{-1}\) taken over all finite subsets \(X\) of \(\mathbb{R}^2\), where \(L_S(X)\) resp. \(L_M(X)\) denotes the total length of a minimum Steiner tree resp. a minimum spanning tree of \(X\). If the length is measured in view of a metric given by a compact, convex, centrally symmetric domain \(D\) instead of the circular unit disc then one gets a Steiner ratio \(\rho(D)\) depending on \(D\). One of the authors (together with others) has conjectured in a previous paper [see Discrete Comput. Geom. 9, No. 4, 351-370 (1993; Zbl 0774.05028)] that \(\rho(D)= \rho(D^*)\) where \(D^*\) denotes the polar body of \(D\). In this paper it is shown that \(\rho_5(D)= \rho_5(D^*)\) with \(\rho_n\) denoting the Steiner ratio taken only of subsets \(X\) consisting of at most five points.