I would like to prove a generalized version of the Minkowski's theorem, but I don't quite know how to do it.
Here is what I would like to prove: Let $X\subset \mathbb{R}^d$ is convex, symmetric around the origin, bounded, and such that ${\rm volume}(X) > k2^d$, then $X$ contains at least $2k$ lattice points.
Thank you for your help.