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

Abstract

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$$ k\ge 3 $$ .