Minkowski's theorem for non-symmetric convex bodies

Question

Minkowski's theorem for convex bodies states that every convex, symmetric subset of $\mathbb{R}^d$ whose volume is larger than $2^d$ contains a non-zero integer point. All the proofs I've seen rely on the symmetric part of it but it seems intuitive that it could be dropped. Are there any results or counterexamples on this?

EDIT: as was pointed out below, if $0$ is not an element of the subset then we can construct a counterexample but what if $0$ is in the set?

Answer 1

Consider the rectangle with vertices at $(-5000,.1),(-5000,.9),(5000,.1),(5000,.9)$. It is a convex subset of ${\bf R}^2$ with volume $8000>2^2$ and no lattice point at all.