The expected size of Heilbronn's triangles

Heilbronn's triangle problem asks for the least a such that n points lying in the unit disc necessarily contain a triangle of area at most Delta. Heilbronn initially conjectured Delta = 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) less than or equal to Delta less than or equal to C/n(8/7-epsilon) for every constant epsilon > 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 theta(1/n(3)); and (ii) the smallest triangle has area theta(1/n(3)) with probability almost one. Our proof uses the incompressibility method based on Kolmogorov complexity.