On the Number of Tetrahedra with Minimum, Unit, and Distinct Volumes in Three-Space

We formulate and give partial answers to several combinatorial problems on volumes of simplices determined bynpoints in 3-space, and in general inddimensions.(i)The number of tetrahedra of minimum (non-zero) volume spanned bynpoints in$\mathbb{R}$3is at most$\frac{2}{3}n^3-O(n^2)$, and there are point sets for which this number is$\frac{3}{16}n^3-O(n^2)$. We also present anO(n3) time algorithm for reporting all tetrahedra of minimum non-zero volume, and thereby extend an algorithm of Edelsbrunner, O'Rourke and Seidel. In general, for every$k,d\in \mathbb{N}, 1\leq k \leq d$, the maximum number ofk-dimensional simplices of minimum (non-zero) volume spanned bynpoints in$\mathbb{R}$dis Θ(nk).(ii)The number of unit volume tetrahedra determined bynpoints in$\mathbb{R}$3isO(n7/2), and there are point sets for which this number is Ω(n3log logn).(iii)For every$d\in \mathbb{N}$, the minimum number of distinct volumes of all full-dimensional simplices determined bynpoints in$\mathbb{R}$d, not all on a hyperplane, is Θ(n).