In Lang - Algebraic number theory, theorem 3, chapter V, page 116, there is a version of Minkowski theorem:
Let $L$ be a lattice of dimension $N$ in $\mathbb R^N$, and let $C$ be a closed, convex, symmetric subset of $\mathbb R^N$. If $\mu(C)\geq 2^N\mu(F)$ where $F$ is a fundamental domain for $L$, there there exists a lattice point $\neq 0$ in $C$.
Here is not required $C$ to be bounded, so I'm some difficulties in proving this: if $(1+\epsilon)C\cap L\neq \{0\}$ for each $\epsilon>0$ then $C\cap L\neq\{0\}$. How to prove this without assuming $C$ bounded (hence compact)?