  We estimate some mixed $L^{p}\left( L^{2}\right) $ 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
$\mathbb{R}^{d}$, $ \left\{ {\int_{\mathbb{T}^{d}}}\left( \frac{1}{H}
{\int_{R}^{R+H}}\left\vert \sum_{k\in\mathbb{Z}^{d}}\chi
_{r\Omega-x}(k)-r^{d}\left\vert \Omega\right\vert
\right\vert^{2}dr\right)^{p/2}dx\right\} ^{1/p}. $ We obtain estimates for
fixed values of $H$ and $R\to\infty$, and also asymptotic estimates when
$H\to\infty$.
