A tight lower bound for the Steiner ratio in Minkowski planes (Q1896344)

A minimum Steiner tree for a given set of points in a metric space is a network interconnecting the points of the set having minimum possible total length. The Steiner ratio for a metric space is the largest lower bound for the ratio of lengths between a minimum Steiner tree and a minimum spanning tree on the same set of points in the space. The present paper shows that for any Minkowski plane (= two-dimensional Banach space) the Steiner ratio is at least \(2/3\). Since the plane with \(L_1\) metric has Steiner ratio \(2/3\) this inequality is best possible.