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 $|{\cal L} \cap K|$.

1. 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. 

2. An easy extension of [Minkowski's convex body theorem][1] 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][2]. 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][3] by Guo for references.


  [1]: https://en.wikipedia.org/wiki/Minkowski%27s_theorem
  [2]: https://en.wikipedia.org/wiki/Gauss_circle_problem
  [3]: https://search.proquest.com/docview/911048762