The width of the transversal strips of \(T\)(3)-families in the plane (Q2572588)

Let \(D\) be a planar convex body. If the width of a strip in a direction is \(w\) times the width of \(D\) in this direction, we say that the strip has relative width \(w\). A family of at least three convex bodies is said to be a \(T(3)\)-family provided every subfamily of three bodies has a line transversal. The author proves that for every planar convex body \(D\) and for every \(w \in (0,1]\) there exists a positive integer \(N(w)\) such that every \(T(3)\)-family of at least \(N(w)\) pairwise disjoint translates of \(D\) has a transversal strip of relative width \(w\). This is a generalization of a result of \textit{J. Eckhoff} [Arch. Math. 24, 195--202 (1973; Zbl 0257.52011)].