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.



Minkowski's theorem for non-0-symmetric sets
**The best reference I know on this question,** restricted to convex polytopes whose vertices are lattice points (symmetry not assumed), is:
Douglas Hensley, *Lattice vertex polytopes with interior lattice points*, Pacific Journal of Mathematics, Vol 101, No. 1, p. 183-191; MR0688412.



> 
> **Author's Abstract.** Consider a convex polytope with lattice vertices and at least one interior lattice point. We prove that the number of boundary lattice
>  points is bounded above by a function of the dimension and the number
>  of interior lattice points. This extends to arbitrary dimension a result of
>  Scott for the two dimensional case.
> 
> 
> 


and



> 
> **An excerpt from the review text in MR:** *The ingenious and elegant
>  proof uses simultaneous Diophantine approximations and some convexity
>  arguments. Especially for k=1, where the results are direct analogues
>  to Minkowski's fundamental theorem on 0-symmetric convex bodies.*
>  (Reviewer: J.M. Wills)
> 
> 
> 


