0 The Average - Case Area of Heilbronn - Type Triangles ∗

triangles with vertices chosen from among n points in the unit square, U , let T be the one with the smallest area, and let A be the area of T . Heilbronn’s triangle problem asks for the maximum value assumed by A over all choices of n points. We consider the average-case: If the n points are chosen independently and at random (uniform distribution) then there exist positive c and C such that c/n < μn < C/n 3 for all large enough n, where μn is the expectation of A. Moreover, with probability close to one c/n < A < C/n. Our proof uses the incompressibility method based on Kolmogorov complexity: it therefore determines the area of the smallest triangle for an arrangement in “general position.”