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)?



Minkowski's theorem in Lang - Algebraic number theory
I claim that either $C$ is bounded or $\mu(C)=\infty$. In either case there is a closed ball $B$ centered at the origin such that $\mu(C\cap B)\geq 2^N\mu(F)$, and then one can apply the argument for $C\cap B$ in place of $C$.


To see the claim, observe first that $C$ is not contained in a hyperplane, hence it contains some closed ball $B(0,r)$ of radius $r>0$ centered at the origin. Let $c\in C\setminus B(0,r)$ be arbitrary. Then $C$ contains the convex hull of $c$ and $B(0,r)$, whose volume is $\gg r^{N-1}|c|$. Hence if $C$ is unbounded, then $\mu(C)=\infty$ as claimed.


