  Let $P$ and $Q$ be polytopes, the first of "low" dimension and the second of
"high" dimension. We show how to triangulate the product $P \times Q$
efficiently (i.e., with few simplices) starting with a given triangulation of
$Q$. Our method has a computational part, where we need to compute an efficient
triangulation of $P \times \Delta^m$, for a (small) natural number $m$ of our
choice. $\Delta^m$ denotes the $m$-simplex.
  Our procedure can be applied to obtain (asymptotically) efficient
triangulations of the cube $I^n$: We decompose $I^n = I^k \times I^{n-k}$, for
a small $k$. Then we recursively assume we have obtained an efficient
triangulation of the second factor and use our method to triangulate the
product. The outcome is that using $k=3$ and $m=2$, we can triangulate $I^n$
with $O(0.816^{n} n!)$ simplices, instead of the $O(0.840^{n} n!)$ achievable
before.
