Depth of segments and circles through points enclosing many points: a note

Neumann-Lara and Urrutia showed in 1985 that in any set of n points in the plane in general position there is always a pair of points such that any circle through them contains at least n−2 60 points. In a series of papers, this result was subsequently improved till n 4.7 , which is currently the best known lower bound. In this paper we propose a new approach to the problem that allows us, by using known results about j-facets of sets of points in R, to give a simple proof of a somehow stronger result: there is always a pair of points such that any circle through them has, both inside and outside, at least n 4.7 points.