Translation cover for closed curves (Q1337816)

Let \({\mathcal F}\) be a family of sets in the plane. A compact, convex set \(T\) is a translation cover for \({\mathcal F}\) if for each set of \({\mathcal F}\) there is a translation that carries the set into \(T\). For special families we may ask for a \(T\) with smallest area. In the paper we prove that a set \(P\) called parabola triangle is a translation cover for the family \({\mathcal K}\) of plane curves of length 1. The area of \(P\) is less than the area of other known translation covers of \({\mathcal K}\) and probably the smallest. We also give a conjecture concerning the translation cover of minimum area for the family of sets of diameter 1.