No selection lemma for empty triangles

Let P be a set of n points in general position in the plane. The Second Selection Lemma states that for any family of \(\Theta(n^3)\) triangles spanned by P, there exists a point of the plane that lies in a constant fraction of them. For families of \(\Theta(n^{3-\alpha})\) triangles, with \(0\le \alpha \le 1\), there might not be a point in more than \(\Theta(n^{3-2\alpha})\) of those triangles. An empty triangle of P is a triangle spanned by P not containing any point of P in its interior. Bárány conjectured that there exists an edge spanned by P that is incident to a super-constant number of empty triangles of P. The number of empty triangles of P might be as low as \(\Theta(n^2)\); in such a case, on average, every edge spanned by P is incident to a constant number of empty triangles. The conjecture of Bárány suggests that for the class of empty triangles the above upper bound might not hold. In this paper we show that, somewhat surprisingly, the above upper bound does in fact hold for empty triangles. Specifically, we show that for any integer n and real number \(0\leq \alpha \leq 1\) there exists a point set of size n with \(\Theta(n^{3-\alpha})\) empty triangles such that any point of the plane is only in \(O(n^{3-2\alpha})\) empty triangles.