Minkowski’s Theorem

Minkowski’s theorem says that if $S$ is a convex set in $\newcommand{\R}{\mathbb{R}}\R^n$ which is symmetric about the origin and has volume $V(S)>2^n$ (or if $S$ is compact and $V(S)\geq2^n$) then $S$ contains some nonzero point of $\newcommand{\x}{\mathbf{x}}\newcommand{\Z}{\mathbb{Z}}\Z^n$.

Minkowski’s theorem can be seen as a simple consequence of Blichfeldt’s theorem. In particular, consider applying its statement to the set $S/2:=\{\,s/2:s\in S\,\}$. Since \[V(S/2)=V(S)/2^n>1\] (or if $S$ is compact, $V(S/2)\geq1$) the theorem says that there exist distinct $\x_1$, $\x_2\in S/2$ with $\x_1-\x_2\in\Z^n$. Say that these have the form $\newcommand{\s}{\mathbf{s}}\x_1=\s_1/2$ and $\x_2=\s_2/2$ where $\s_1$, $\s_2\in S$.

Since $S$ is symmetric about the origin, it follows that $-\s_2\in S$. Additionally, since $S$ is convex, it follows that the midpoint of $\s_1$ and $-\s_2$ is also in $S$. But this midpoint $(\s_1-\s_2)/2=\x_1-\x_2$ is also in $\Z^n$. Since $\x_1\neq\x_2$ this point is nonzero, as required.

Using the form of Blichfeldt’s theorem applied to general lattices $L$ of dimension $n$, one finds that if $S$ is a convex and symmetric set in $\R^n$ with volume $V(S)>2^n\det(L)$ (or if $S$ is compact and $V(S)\geq2^n\det(L)$) then $S$ contains a nonzero point of $L$.