On the number of tetrahedra with minimum, unit, and distinct volumes in three-space

• Authors: Adrian Dumitrescu;Csaba d. TÓth
• Venue: Combinatorics, Probability and Computing
• Year: 2008

We formulate and give partial answers to several combinatorial problems on volumes of simplices determined by n points in 3-space, and in general in d dimensions. (i)The number of tetrahedra of minimum (non-zero) volume spanned by n points in $\mathbb{R}$3 is 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 an O(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 of k-dimensional simplices of minimum (non-zero) volume spanned by n points in $\mathbb{R}$d is Θ(nk). (ii)The number of unit volume tetrahedra determined by n points in $\mathbb{R}$3 is O(n7/2), and there are point sets for which this number is Ω(n3 log logn). (iii)For every $d\in \mathbb{N}$, the minimum number of distinct volumes of all full-dimensional simplices determined by n points in $\mathbb{R}$d, not all on a hyperplane, is Θ(n).