The Katchalski-Lewis transversal problem for regular polygons
Let \(\mathcal F\) be a family of plane convex bodies. We say that: (i) \(\mathcal F\) has property \(T(k)\) if \(| \mathcal F| \geq k\) and any of its subfamilies with \(k\) members has a line transversal; (ii) \(\mathcal F\) has property \(T-m\) if there exists a subfamily of \(\mathcal F\) with \(|\mathcal F \setminus \mathcal G| \leq m\) which has a line transversal. The main result of the paper is the following: Let \(\mathcal F\) be a finite \(T(3)\)-family of disjoint translates of a regular \(2n\)-gon, where \(n \geq 5\). The following statements hold: (i) \(\mathcal F\) has property \(T-3\) for \(5 \leq n \leq 34\); (ii) \(\mathcal F\) has property \(T-2\) for \(n \geq 34\).
scientific article; zbMATH DE number 6909529
scientific article