The Steiner minimal network for convex configurations (Q1207800)

If \(X\) is a finite set of points in the Euclidean plane, the Steiner problem is to find a shortest network connecting the points. In general, this problem is NP-hard. In a former paper by \textit{D. A. Thomas} and \textit{J. H. Rubinstein} [ibid. 7, No. 1, 77-86 (1992; see the review above)] it is shown that the Steiner problem is much easier, if \(X\) lies on a circle. Generalizing this result the paper shows: Suppose \(X\) is a convex configuration with radius of maximum curvature \(r\) and at most one of the edges joining neighboring points has length strictly greater than \(r\), then the shortest network (the Steiner tree) consists of all the edges with a longest edge removed.