Steiner minimal trees on sets of four points
Let \(S=\{A,B,C,D\}\) consist of the four corner points of a convex quadrilateral where diagonals [A,C] and [B,D] intersect at the point O. There are two possible full Steiner trees for S, the AB-CD tree has A and B adjacent to one Steiner point, and C and D to another; the AD-BC tree has A and D adjacent to one Steiner point, and B and C to another. Pollak proved that if both full Steiner trees exist, then the AB-CD (AD-BC) tree is the Steiner minimal tree if \(\sphericalangle AOD>(<)90\circ\), and both are Steiner minimal trees if \(\sphericalangle AOD=90\circ\). While the theorem has been crucially used in obtaining results on Steiner minimal trees in general, its applicability is sometimes restricted because of the condition that both full Steiner trees must exist. In this paper we remove this obstacle by showing: (i) Necessary and sufficient conditions for the existence of either full Steiner tree for S. (ii) If \(\sphericalangle AOD\geq 90\circ\), then the AB-CD tree is the SMT even if the AD-BC tree does not exist. (iii) If \(\sphericalangle AOD<90\circ\) but the AD-BC tree does not exist, then the AB-CD tree cannot be ruled out as a Steiner minimal tree, though under certain broad conditions it can.
scientific article; zbMATH DE number 4010535