Let $\cal L$ be a full-dimensional lattice in $\mathbb{R}^n$ with determinant $\det({\cal L})$. Let $V$ be the Voronoi cell of $\cal L$, i.e. the set of all points in $\mathbb{R}^n$ which are closer to $0$ (in Euclidean distance) than to any other point of $\cal L$. Let $K$ be a convex body symmetric around $0$. We have the following volumetric bounds on $|{\cal L} \cap K|$.
- Because $({\cal L} \cap K) + V \subseteq K + V$ and ${\cal L} + V$ is a packing (i.e. for any two distinct lattice points $x$ and $y$ $(x + V) \cap (y + V) = \emptyset$), we have
$$ |{\cal L} \cap K| \le \frac{\mathrm{vol}(K + V)}{\mathrm{vol}(V)} = \frac{\mathrm{vol}(K + V)}{\det({\cal L})}. $$ This works with $V$ replaced by any other set that tiles space with respect to $\cal L$, e.g. any fundamental parallelepiped.
- An easy extension of Minkowski's convex body theorem shows that
$$ |{\cal L} \cap K| \ge \frac{\mathrm{vol}(K)}{2^n\det({\cal L})}. $$
This problem is also studied in a setting analogous to the Gauss circle problem. Let's just look at the case ${\cal L} = \mathbb{Z}^n$ (you can always reduce to this case by applying a linear transformation to both $K$ and $\cal L$). Define the discrepancy function $D_K(t) = |tK \cap \mathbb{Z}^n| - t^n \mathrm{vol}(K)$. It's a long standing open problem to find the smallest $c$ so that $|D_K(t)| = O(t^{d-2 + c})$. Here the constant in the asymptotic notation could depend on $K$. Check this thesis by Guo for references.