Generalizations of Heilbronn’s triangle problem

For given integers d , j ≥ 2 and any positive integers n , distributions of n points in the d -dimensional unit cube [ 0 , 1 ] d are investigated, where the minimum volume of the convex hull determined by j of these n points is large. In particular, for fixed integers d , k ≥ 2 the existence of a configuration of n points in [ 0 , 1 ] d is shown, such that, simultaneously for j = 2 , … , k , the volume of the convex hull of any j points among these n points is Ω ( 1 / n ( j − 1 ) / ( 1 + | d − j + 1 | ) ) . Moreover, a deterministic algorithm is given achieving this lower bound, provided that d + 1 ≤ j ≤ k .