On five points in the boundary of a plane convex body pairwise in at least unit relative distances (Q1895164)

For a compact convex set \(C\) with nonempty interior, the \(C\)-distance of points \(a,b\) is two times the ratio of their Euclidean distance and the distance of \(a'\) and \(b'\), both in \(C\), such that \(ab\) and \(a'b'\) are parallel. It is known that there are always five points in \(C\) in at least unit relative distance. This short paper shows that these points can be chosen on the boundary of \(C\).