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  "body": "\n## Statement\n\nLet $n$ be a positive integer, and let $L \\subset \\mathbb{R}^n$ be a lattice with determinant $d(L)$.\nLet $S \\subseteq \\mathbb{R}^n$ be a convex set that is symmetric about the origin: $x \\in S$ implies $-x \\in S$.\nIf the volume of $S$ is greater than $2^n d(L)$, then $S$ contains a point of $L$ other than $0$.\n\n## Remarks\n\nThe most common case is $L = \\mathbb{Z}^n$, with $d(L) = 1$: a symmetric convex set of volume greater than $2^n$ contains a nonzero integer point.\nThe bound is sharp: the open cube $(-1, 1)^n$ has volume $2^n$ and contains no nonzero integer point.\nApplications include proofs of [[Fermat's theorem on sums of two squares|043B3ENSTY]] and [[Lagrange's four-square theorem|0KYYSMEVHH]].\n\n## History\n\nHermann Minkowski proved the theorem in 1889. It started the geometry of numbers.\n\n## Explanations\n\n### L1 human\n\nA large, round, centered region in the plane must contain a lattice point other than the origin.\nIn the plane with $L = \\mathbb{Z}^2$, the area must be more than $4$.\n",
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