Distributions of points in the unit-square and large k-gons

We consider a generalization of Heilbronn's triangle problem by asking, given any integers <i>n</i> ≥ <i>k</i> ≥ 3, for the supremum Δ<inf><i>k</i></inf>(<i>n</i>) of the minimum area determined by the convex hull of some <i>k</i> of <i>n</i> points in the unit-square [0, 1]², where the supremum is taken over all distributions of <i>n</i> points in [0, 1]². Improving the lower bound Δ<inf><i>k</i></inf>(<i>n</i>) = Ω(1/n^{(<i>k</i>-1)/(<i>k</i>-2}}) from [5] and from [20] for <i>k</i> = 4, we will show that Δ<inf><i>k</i></inf>(<i>n</i>) = Ω((log <i>n</i>)^{1/(<i>k</i>-2)}/<i>n</i>^{(<i>k</i>-1)/(<i>k</i>-2)} for each fixed integer <i>k</i> ≥ 3 as asked for in [5]. We will also provide a deterministic polynomial time algorithm which finds <i>n</i> points in the unit-square [0, 1]² achieving this lower bound.