The maximum size of a convex polygon in a restricted set of points in the plane

Assume we have k points in general position in the plane such that the ratio between the maximum distance of any pair of points to the minimum distance of any pair of points is at most α√k, for some positive constant α. We show that there exist at least βk^1/4 of these points which are the vertices of a convex polygon, for some positive constant β=β(α). On the other hand, we show that for every fixed ε>0, if k>k(ε), then there is a set of k points in the plane for which the above ratio is at most 4√k, which does not contain a convex polygon of more than k^1/3+ε vertices.