A question from a famous paper of Erdős (Q2391829)

Paul Erdős made once a claim that ``On every convex curve there exists a point \(P\) such that every circle with centre \(P\) intersects the curve in at most 2 points.'' This claim is known to be false. The paper investigates a modified problem: Given a convex domain \(K\) in the plane, what is the smallest number \(N\), such that there is point \(P\) on the boundary of the domain, such that any circle centered at \(P\) intersects the boundary of \(K\) in at most \(N\) points. \(N=2\) fails as noted above. The paper concludes that no \(K\) has \(N=\infty\) and that there are \(K\)'s with \(N=6\).