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 $\\newcommand{\\x}{\\mathbf{x}}\\newcommand{\\Z}{\\mathbb{Z}}\\Z^n$.<\/p>\n
Minkowski’s theorem can be seen as a simple consequence of Blichfeldt’s theorem<\/a>. In particular, consider applying its statement<\/a> 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 $\\s_1$, $\\s_2\\in S$.<\/p>\n 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\u00a0$(\\s_1-\\s_2)\/2=\\x_1-\\x_2$ is also in $\\Z^n$. Since $\\x_1\\neq\\x_2$ this point is nonzero, as required.<\/p>\n 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 $L$.<\/p>\n","protected":false},"excerpt":{"rendered":" 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 … Continue reading