Suppose $K$ is a convex set in $\mathbb R^n$ which is symmetric with respect to the origin. Minkowski's theorem tells us that if the volume of $K$ is greater than $2^n$, then $K$ contains a nonzero integer point.
According to this MO post, it is claimed that $ |\mathbb Z^n \cap K| \ge \frac{1}{2^n}\mathrm{vol}(K) $ and the bound is in general tight. Can someone provide some reference on this? Thank you!