On a Helly-type question for central symmetry (Q2322535)

In 2010, K. Swanewpoel posed the following Helly-type problem: determine the existence of \(k\in{\mathbb{N}}\) such that, for any planar set \(X\), if any \(k\) points of \(X\) are in centrally symmetric convex (c.s.c.) position, then the whole set \(X\) is so. Here, it is said that a set is in c.s.c. position if it is contained in the boundary of a centrally symmetric convex body. This question is known to be true for \(k\leq 5\).\par In the paper under review, the authors prove that the above problem has not a positive solution when \(k=8\). They moreover show that, if \(X\) is a simple planar closed convex curve, then the answer is affirmative for \(k=6\).