Geometric permutations and common transversals
Let \({\mathcal F}\) be a finite family of disjoint convex subsets of the plane and p(\({\mathcal F})\) the family of all geometric permutations of \({\mathcal F}\). For the notion of geometric permutation and other results see the paper of the first two authors and \textit{J. Zaks} [Discrete Math. 54, 271-284 (1985; Zbl 0561.52002). In this paper the authors give the following two results: (1) n-1 \(\leq \max | p({\mathcal F})| \leq \left( \begin{matrix} n\\ 2\end{matrix} \right)\), where the maximum is taken over all families of n pairwise disjoint convex sets. (2) \(| p({\mathcal F})| \leq 8\), for every family \({\mathcal F}\) of at least 11 pairwise disjoint translates of a convex set. The authors notice that in the paper mentioned above the lower bound in (1) has been improved to 2n-2.
family of disjoint convex subsets of the plane
scientific article; zbMATH DE number 3979919