The expected size of Heilbronn´s triangles

Heilbronn´s triangle problem asks for the least Δ such that n points lying in the unit disc necessarily contain a triangle of area at most Δ. Heilbronn initially conjectured Δ=O(1/n² ). As a result of concerted mathematical effort it is currently known that there are positive constants c and C such that c log n/n²⩽Δ⩽C/n^8/7-ε for every constant ε>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/n³); and (ii) the smallest triangle has area Θ(1/n³) with probability almost one. Our proof uses the incompressibility method based on Kolmogorov complexity