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 points in the unit square [ 0, 1] 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 and a set. of n points in [0, 1](2), let A(k)(rho) be the minimum area of the convex hull of k points in.. Here, instead of considering the supremum of Ak(n) over all such choices of., we consider its average value, .k(n), when the n points in. are chosen independently and uniformly at random in [ 0, 1] 2. We prove that Delta k(n) = T (n -k k- 2), for every fixed k >= 3.