On seven points in the boundary of a plane convex body in large relative distances (Q1875922)

In 1991 Lassak raised the problem of finding configurations of points in the boundary of a convex body \(C\) which are far in the sense of the following notion of \(C\)-distance of points: For arbitrary points \(p,q\) in the Euclidean \(n\)-space, let \(| pq |\) denote the Euclidean length of the segment \(pq\). Let \(p'q'\) be a chord in \(C\) parallel to \(pq\) such that there is no longer chord of \(C\) parallel to \(pq\). The \(C\)-distance \(d_C(p,q)\) of points \(p\) and \(q\) is defined by the ratio of \(| pq |\) to \(\frac 12 | p'q' |\). If there is no doubt about \(C\) the term relative distance of \(p\) and \(q\) is used. In the paper it is proved that the boundary of an arbitrary planar convex body contains seven points in pairwise relative distances at least \(\frac 23\) such that the relative distances of every two successive points are equal. The example of a triangle shows that the estimate \(\frac 23\) cannot be improved. The author also improves the estimate \(\frac 43\) obtained by Bezdek, Fodor and Talata in 1991 for three points in the boundary of a planar convex body. Several consequences about configurations of other numbers of points and also about packings by homothetical copies of a planar convex body are deduced from the main results.