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 2/3n3−O(n2), and there are point sets
for which this number is 3/16n3−O(n2). We also present an O(n3) 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≤k≤d, the maximum number of k-dimensional
simplices of minimum (nonzero) volume spanned by n points in \RR^d is
Θ(nk). (ii) The number of unit-volume tetrahedra determined by n
points in \RR^3 is O(n7/2), and there are point sets for which this
number is Ω(n3loglogn). (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 Θ(n).Comment: 19 pages, 3 figures, a preliminary version has appeard in proceedings
of the ACM-SIAM Symposium on Discrete Algorithms, 200