  Let $X$ be a Poisson point process and $K\subset\mathbb{R}^d$ a measurable
set. Construct the Voronoi cells of all points $x\in X$ with respect to $X$,
and denote by $v_X(K)$ the union of all Voronoi cells with nucleus in $K$. For
$K$ a compact convex set the expectation of the volume difference
$V(v_X(K))-V(K)$ and the symmetric difference $V(v_X(K)\triangle K)$ is
computed. Precise estimates for the variance of both quantities are obtained
which follow from a new jackknife inequality for the variance of functionals of
a Poisson point process. Concentration inequalities for both quantities are
proved using Azuma's inequality.
