Note on the number of obtuse angles in point sets

Abstract

In 1979 Conway, Croft, Erd\H{o}s and Guy proved that every set SS of nn points in general position in the plane determines at least n3/18−O(n2) obtuse angles and also presented a special set of nn points to show the upper bound 2n3/27−O(n2) on the minimum number of obtuse angles among all sets SS. We prove that every set SS of nn points in convex position determines at least 2n327−o(n3)2n327−o(n3) obtuse angles, hence matching the upper bound (up to sub-cubic terms) in this case. Also on the other side, for point sets with low rectilinear crossing number, the lower bound on the minimum number of obtuse angles is improved.