We define a lattice $\Lambda$ in $\mathbb{R}^n$ for a given basis $u_{1},\dots, u_{n}$ as the set $\{ a_{1} u_{1} + \dots + a_{n}u_{n} \mid a_{i} \in \mathbb{Z} \}.$ > [!NOTE] Minkowski's theorem > Let $S$ be a symmetric, convex set with $\text{vol}(S)>2^n \det \Lambda$. Then there exists $\bar{x}\in \Lambda \cap S \setminus \{ 0 \}.$ > > [!NOTE] Strong Form of Minkowski's theorem > Let $S$ be a [[Compact topological spaces|compact]], symmetric, convex set with $\text{vol}(S) \geq 2^n \det \Lambda$. Then there exists $\bar{x}\in \Lambda \cap S \setminus \{ 0 \}$. ###### Proof ... ###### Proof of Strong Form of Minkowski's theorem For $k\in \mathbb{N}$, define $S_{k}=\left( 1+\frac{1}{k} \right)S$. Then each $S_{k}$ has volume exceeding $2^n \det \Lambda$ and is symmetric, convex. Thus by the above theorem, each intersection $S_{k} \cap \Lambda \setminus \{ 0 \}$ is non-empty. These sets are compact so, by [[Cantor's Intersection Theorem]] $S_{k} \cap \Lambda \setminus \{ 0 \}$is non-empty.