Integer points in parallelepiped

Question

There is a parallelepiped in $n$ dimensional vector space over $\mathbb{R}$. All its vertex are integer. Its volume $V>1$. How to prove that there is an integer point which belongs to the parallelepiped but it is not a vertex?

Answer 1

Given a parallelepiped $P$ in $\mathbb R^n$ with all coordinates integers, scale $P$ by a factor of $2$ and centre the result on the origin, giving $Q$. Since $P$ has volume greater than $1$, $Q$ has volume greater than $2^n$. $Q$ is also symmetric about the origin and convex, so Minkowski's theorem guarantees at least one interior non-origin point in $Q$ and therefore $P$.