Distributions of Points and Large Quadrangles

Abstract

Abstract. We consider a variant of Heilbronn's triangle problem by asking, given any integer n ≥ 4, for the supremum ∆4(n) of the minimum area determined by the convex hull of some four of n points in the unit-square [0, 1] 2 over all distributions of n points in [0, 1] 2 . Improving the lower bound ∆4(n) = Ω(1/n 3/2 ) of Schmidt [19], we will show that ∆4(n) = Ω((log n) 1/2 /n 3/2 ) as asked for in [5], by providing a deterministic polynomial time algorithm which finds n points in the unit-square [0, 1] 2 that achieve this lower bound.