AN UPPER BOUND FOR THE d-DIMENSIONAL ANALOGUE OF HEILBRONN'S TRIANGLE PROBLEM

摘要

In this paper it is shown that for any set of n points selected from the d-dimensional unit cube, d odd, the volume of the smallest simplex spanned by the set is O(n~(-(1 + 1/(2d))), which is a slight improvement on the only known upper bound O(n~(-1)), although still far from the lower bound Ω(n~(-d) log n).