A Lower Bound for Heilbronn's Triangle Problem in d Dimensions

Abstract

In this paper we show a lower bound for the generalization of Heilbronn's triangle problem to d dimensions; namely, we show that there exists a set S of n points in the d-dimensional unit cube so that every d+1 points of S define a simplex of volume $\Omega (\frac{1}{n^d})$. We also show a constructive incremental positioning of n points in a unit 3-cube for which every tetrahedron defined by four of these points has volume $\Omega (\frac{1}{n^4})$.