On circles containing the maximum number of points (Q1916370)

Let \(B(x,y)\) be the disk in the plane \(\mathbb{R}^2\) which has points \(x\), \(y\) as its diametral end points. And let \(\Pi^B(n)\) (or \(\Pi^B(n)\)) be the largest number such that for every set (or every convex set) \(P\) of \(n\) \((n \geq 2)\) points in \(\mathbb{R}^2\), there exist two points \(x, y \in P\) for which \(B(x,y)\) contains \(\Pi^B(n)\) (or \(\overline{\Pi}^B(n)\)) points of \(P\). In this note is proved that \(\Pi^B (n) = \overline{\Pi}^B(n) = \lceil n/3\rceil + 1\). This investigation is related to functions \(\Pi(n)\) and \(\overline{\Pi}(n)\) which are studied by \textit{V. Neumann-Lara} and \textit{J. Urrutia} [Discrete Math. 69, No. 2, 113-178 (1988; Zbl 0645.05024)] and \textit{R. Hayword} [Discrete Comput. Geom. 4, No. 3, 263-264 (1989; Zbl 0673.52008)].