  We estimate the $L^{p}$ norms of the discrepancy between the volume and the
number of integer points in $r\Omega-x$, a dilated by a factor $r$ and
translated by a vector $x$ of a convex body $\Omega$ in $\mathbb{R}^{d}$ with
smooth boundary with strictly positive curvature, \[ \left\{
{\displaystyle\int_{\mathbb R}}{\displaystyle\int_{\mathbb{T}^{d}}}\left\vert
\sum_{k\in\mathbb{Z}^{d}}\chi _{r\Omega-x}(k)-r^{d}\left\vert \Omega\right\vert
\right\vert ^{p}dxd\mu(r-R) \right\} ^{1/p}, \] where $\mu$ is a Borel measure
compactly supported on the positive real axis and $R\to+\infty$.
