Can you help proving this generalization of the Minkowski theorem ?
Let $\Omega \subset \mathbb{R}^N$ be a nonempty centrally symmetric convex subset.
a) Then $\#(\Omega \cap \mathbb{Z}^N) \ge 2 (\lceil \frac{Vol(\Omega)}{2^N}\rceil-1)+1$
b) If $\Omega$ is closed and bounded, then $\#(\Omega \cap \mathbb{Z}^N) \ge 2 \lfloor \frac{Vol(\Omega)}{2^N}\rfloor+1$
I found the answer in here: http://www.fmf.uni-lj.si/~lavric/Shmonin%20-%20Minkowski%27s%20theorem%20and%20its%20applications.pdf
The idea is just instead of 2 vectors with difference that is an integer vector we must have $m$ of those (where the volume is $m*2^n$)