Abstract

Heilbronn’s triangle problem asks for the least ∆ such that n points lying in the unit disc necessarily contain a triangle of area at most ∆. Heilbronn initially conjectured ∆ = O(1/n 2). As a result of concerted mathematical effort it is currently known that there are positive constants c and C such that c log n/n 2 ≤ ∆ ≤ C/n 8/7−ǫ for every constant ǫ> 0. We resolve Heilbronn’s problem in the expected case: If we uniformly at random put n points in the unit disc then (i) the area of the smallest triangle has expectation Θ(1/n 3); and (ii) the smallest triangle has area Θ(1/n 3) with probability almost one. Our proof uses the incompressibility method based on Kolmogorov complexity. 1

Citations