Heilbronn conjectured that given arbitrary n points in the 2-dimensional unit square [0, 1]², there must be three points which form a triangle of area at most O(1/n²). This conjecture was disproved by a nonconstructive argument of Komlós, Pintz and Szemerédi [10] who showed that for every n there is a configuration of n points in the unit square [0, 1]² where all triangles have area at least (log n/n²). Considering a generalization of this problem to dimensions d3, Barequet [3] showed for every n the existence of n points in the d-dimensional unit cube [0, 1]^d such that the minimum volume of every simplex spanned by any (d+1) of these n points is at least (1/n^d). We improve on this lower bound by a logarithmic factor (log n).