On stable line segments in all triangulations of a planar point set (Q1920445)

Let \(S\) be a finite set of points of the euclidean plane such that no three of them are collinear. A line segment \(L(p,q)\) joining two points \(p,q \in S\) is called stable, if no other line segment with endpoint in \(S\) intersects \(L(p,q)\). A triangulation of \(S\) is a maximal set of non-intersecting line segments between points of \(S\). The author proves that the set \(SL (S)\) of all stable line segments of \(S\) is just the intersection of all triangulations of \(S\). Moreover, he shows that the maximum number \(M(n)\) of stable lines of a set \(S\) of cardinality \(n\) is always \(2(n-1)\), if \(n\geq 4\).