The expected size of Heilbronn's triangles

Abstract

Heilbronn's triangle problem asks for the least /spl Delta/ such that n points lying in the unit disc necessarily contain a triangle of area at most /spl Delta/. Heilbronn initially conjectured /spl Delta/=O(1/n/sup 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/sup 2//spl les//spl Delta//spl les/C/n/sup 8/7-/spl epsiv// for every constant /spl epsiv/>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 /spl Theta/(1/n/sup 3/); and (ii) the smallest triangle has area /spl Theta/(1/n/sup 3/) with probability almost one. Our proof uses the incompressibility method based on Kolmogorov complexity.