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

TL;DR

This paper investigates the combinatorial properties of tetrahedra and simplices determined by points in three and higher dimensions, providing bounds, algorithms, and asymptotic results for volumes and counts.

Contribution

It establishes bounds on the number of minimum and unit-volume tetrahedra, presents an algorithm for reporting minimum volume tetrahedra, and analyzes the minimum number of distinct volumes in high dimensions.

Findings

Maximum number of minimum-volume tetrahedra is proportional to n^3.

Number of unit-volume tetrahedra is O(n^{7/2}) with some point sets reaching Omega(n^3 log log n).

Minimum number of distinct volumes of simplices is proportional to n.

Abstract

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 (nonzero) volume spanned by $n$ points in $\RR^3$ is at most ${2/3}n^3-O(n^2)$, and there are point sets for which this number is ${3/16}n^3-O(n^2)$. We also present an $O(n^3)$ time algorithm for reporting all tetrahedra of minimum nonzero volume, and thereby extend an algorithm of Edelsbrunner, O'Rourke, and Seidel. In general, for every $k,d\in \NN$, $1\leq k \leq d$, the maximum number of $k$-dimensional simplices of minimum (nonzero) volume spanned by $n$ points in $\RR^d$ is $\Theta(n^k)$. (ii) The number of unit-volume tetrahedra determined by $n$ points in $\RR^3$ is $O(n^{7/2})$, and there are point sets for which this number is $\Omega(n^3 \log \log{n})$. (iii) For every $d\in \NN$, the minimum number of distinct volumes of all full-dimensional simplices determined by $n$ points in $\RR^d$, not all on a hyperplane, is $\Theta(n)$.