The maximum size of a convex polygon in a restricted set of points in the plane
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Assume we havek 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 mostα√k, for some positive constantα. We show that there exist at leastβk1/4 of these points which are the vertices of a convex polygon, for some positive constantβ=β(α). On the other hand, we show that for every fixedε>0, ifk>k(ε), then there is a set ofk points in the plane for which the above ratio is at most 4√k, which does not contain a convex polygon of more thank1/3+ε vertices.
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