Heilbronn triangle‐type problems in the unit square [0,1]2

The Heilbronn triangle problem is a classical geometrical problem that asks for a placement of n$$ n $$ points in the unit square [0,1]2$$ {\left[0,1\right]}^2 $$ that maximizes the smallest area of a triangle formed by three of those points. This problem has natural generalizations. For an integer k≥3$$ k\ge 3 $$ and a set P of n$$ n $$ points in [0,1]2$$ {\left[0,1\right]}^2 $$, let Ak(P) be the minimum area of the convex hull of k$$ k $$ points in P. Here, instead of considering the supremum of Ak(P) over all such choices of P, we consider its average value, Δ ̃k(n)$$ {\tilde{\Delta}}_k(n) $$, when the n$$ n $$ points in P are chosen independently and uniformly at random in [0,1]2$$ {\left[0,1\right]}^2 $$. We prove that Δ ̃k(n)=Θn−kk−2$$ {\tilde{\Delta}}_k(n)=\Theta \left({n}^{\frac{-k}{k-2}}\right) $$, for every fixed k≥3$$ k\ge 3 $$.