Minkowski's Convex Body Theorem

Theorem (Minkowski's Convex Body Theorem): Let $S \subseteq \mathbb{R}^n$. If $S$ is convex, symmetric about the origin, and $\mathrm{volume}(S) > 2^n$ then $S$ contains a nonzero integer point.

Each of the conditions in Minkowski's convex body theorem are necessary.

For example, the inequality in Minkowski's convex body theorem is strict. For example, if $S = (-1, 1) \times (-1, 1) \subset \mathbb{R}^2$ then $S$ is convex, symmetric about the origin, and $\mathrm{volume}(S) = 2^2 = 4$, but $S$ contains no nonzero integer point.

We now state a generalized version of Minkowski's convex body theorem.

Theorem 2 (Minkowski's Generalized Convex Body Theorem): Let $\Lambda = A \mathbb{Z}^n$ be a lattice and let $S \subseteq \mathbb{R}^n$. If $S$ is convex, symmetric about the origin, and $\mathrm{volume} (S) > 2^n d(\Lambda)$ then $S$ contains a nonzero lattice point of $\Lambda$.

Recall that $d(\Lambda)$ denotes the determinant of the lattice $\Lambda$ and is defined to be $d(\Lambda) = |\det(A)|$.