Partitions of planar sets into small triangles

Given 3n points in the unit square, n ≥ 2, they determine n triangles whose vertices exhaust the given 3n points in many ways. Choose the n triangles so that the sum of their areas is minimal, and let a^*(n) be the maximum value of this minimum over all configurations of 3n points. Then n^{- 1 2}≪a^*(n)≪ n^{- 1 9} is deduced using results on the Heilbronn triangle problem. If the triangles are required to be area disjoint it is not even clear that the sum of their areas tends to zero; this open question is examined in a slightly more general setting.