On circles containing the maximum number of points

Abstract 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) [or Π B (n)] be the largest number such that for every set [or every convex set] P of n points in R 2 , there exist two points x, y ϵ P for which B(x, y) contains Π B (n) [ or Π B (n)] points of P. We show that Π B (n) = Π B (n) = ⌈n/3⌉ + 1 .