  For a convex body $K \subset {\mathbb R}^n$, let $K^z = \{y\in{\mathbb R}^n :
\langle y-z, x-z\rangle\le 1, \mbox{\ for all\ } x\in K\}$ be the polar body of
$K$ with respect to the center of polarity $z \in {\mathbb R}^n$. The goal of
this paper is to study the maximum of the volume product
$\mathcal{P}(K)=\min_{z\in {\rm int}(K)}|K||K^z|$, among convex polytopes
$K\subset {\mathbb R}^n$ with a number of vertices bounded by some fixed
integer $m \ge n+1$. In particular, we prove that the supremum is reached at a
simplicial polytope with exactly $m$ vertices and we provide a new proof of a
result of Meyer and Reisner showing that, in the plane, the regular polygon has
maximal volume product among all polygons with at most $m$ vertices. Finally,
we treat the case of polytopes with $n+2$ vertices in ${\mathbb R}^n$.
