Large Triangles in the d-Dimensional Unit-Cube

We consider a variant of Heilbronn’s triangle problem by asking for fixed dimension d ≥ 2 for a distribution of n points in the d-dimensional unit-cube [0, 1] such that the minimum (2-dimensional) area of a triangle among these n points is maximal. Denoting this maximum value by Δoff−line d (n) and Δ on−line d (n) for the off-line and the online situation, respectively, we will show that c1 ·(log n)1/(d−1)/n2/(d−1) ≤ Δoff−line d (n) ≤ C1/n and c2/n2/(d−1) ≤ Δon−line d (n) ≤ C2/n for constants c1, c2, C1, C2 > 0 which depend on d only.