Let $\Lambda \subseteq \mathbb{R}^n$ be a full-rank lattice, i.e. $\Lambda = A \mathbb{Z}^n$ for some $A \in \mathrm{GL}_n (\mathbb{R})$, and let $C \subseteq \mathbb{R}^n$ be a $0$-symmetric convex body. Then Minkowski's theorem asserts that $$ \# | \Lambda \cap C | \geq \frac{|C|}{2^n | \mathbb{R}^n / \Lambda|}. $$ I was curious if there is a version of Minkowski's theorem that holds for convex bodies not necessarily symmetric around $0$. Clearly, the body should contain $0$, otherwise we might have $\# | \Lambda \cap C | = 0$. Are there other conditions on $C$ than symmetry around $0$ that still allow for a lower bound on the number of lattice points in $C$?

Any comment or reference is highly appreciated.