I've been doing some reading around the number of points of $\mathbb{Z}^n$ inside an arbitrary rank $n$, $0$-symmetric convex body $K$. In particular, I came across Blichfeldt's remarkable bound:
$$ \left|K \cap \mathbb{Z}^n\right| \leq n!\text{Vol}(K)+n. $$
However, I'm struggling to find any literature regarding lower bounds on the value of $\left|K \cap \mathbb{Z}^n\right|$. I'm wondering if any such lower bounds exists in literature, in particular when the lower bound is given in terms of the volume of $K$ (as Blichfeldt's upper bound is given)? Assuming of course that the volume of $K$ is large enough so that $K \cap \mathbb{Z}^n$ contains at least one nonzero lattice point.