On circles containing the maximum number of points Original Research Article

We define B(x, y) to be the disk in the plane which has the points x, y as its diametral end points. Let ΠB(n) [orΠB(n)] be the largest number such that for every set [or every convex set] P of n points in R2, there exist two points x, y ϵ P for which B(x, y) contains ΠB(n) [orΠB(n)] points of P. We show that ΠB(n) = ΠB(n) = ⌈n/3⌉ + 1.