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

Abstract

AbstractThe Heilbronn triangle problem is a classical geometrical problem that asks for a placement of points in the unit square that maximizes the smallest area of a triangle formed by three of those points. This problem has natural generalizations. For an integer and a set of points in , let be the minimum area of the convex hull of points in . Here, instead of considering the supremum of over all such choices of , we consider its average value, , when the points in are chosen independently and uniformly at random in . We prove that , for every fixed .