Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 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 $\Theta(n)$.

10
The categories of this paper are: q-bio
5
The categories of this paper are: gr-qc
0
The categories of this paper are: hep-ph
3
The categories of this paper are: physics
20
The categories of this paper are: acc-phys
27
The categories of this paper are: comp-gas
15
The categories of this paper are: nucl-ex
6
The categories of this paper are: astro-ph
24
The categories of this paper are: dg-ga
32
The categories of this paper are: atom-ph
18
The categories of this paper are: econ
8
The categories of this paper are: hep-ex
37
The categories of this paper are: supr-con
13
The categories of this paper are: quant-ph
34
The categories of this paper are: chem-ph
12
The categories of this paper are: nucl-th