Proof of Grünbaum's conjecture on common transversals for translates (Q1262538)

Let F be a family of pairwise disjoint translates of a compact convex set K in the plane. One says that F has property T(k) if any k sets from F have a common transversal. Grünbaum conjectured that, if F has property T(5) and consists of at least 5 sets, then there is a common transversal for all sets in F and \(k=5\) is best possible [\textit{B. Grünbaum}, Arch. Math. 9, 465-469 (1958; Zbl 0083.172)]. In this paper the author gives a proof of the conjecture, based on successive and delicate steps for excluding (by hand and by computer) myriads of combinations of 6 partial transversals, remaining only two possibilities which are analyzed directly to prove that they cannot arise.