Abstract

Let be a set of points in . Bennett et al. (2016) proved that if then determines a positive proportion of all -simplices. In this paper, we give an improvement of this result in the case when is the Cartesian product of sets. Namely, we show that if is the Cartesian product of sets and , the number of congruence classes of -simplices determined by is at least , and in some cases our result is sharp.