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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $\\Lambda \\subset \\R^n$ be a [[lattice|WN9V7WSJKF]] with covolume $\\det \\Lambda$.\nLet $K \\subset \\R^n$ be a [[convex|02AS45T3H9]] [[compact|0R6EWFKRCF]] set with nonempty interior that is symmetric about the origin, that is, $-x \\in K$ for all $x \\in K$.\nFor $k = 1, \\dots, n$, the $k$-th successive minimum $\\lambda_k$ is the infimum of the real numbers $\\lambda \u003e 0$ such that $\\lambda K$ contains $k$ [[linearly independent|0CRMGPAH83]] vectors of $\\Lambda$.\nThen $0 \u003c \\lambda_1 \\le \\lambda_2 \\le \\dots \\le \\lambda_n \u003c \\infty$ and\n$$\\frac{2^n}{n!} \\det \\Lambda \\le \\lambda_1 \\lambda_2 \\cdots \\lambda_n \\operatorname{vol}(K) \\le 2^n \\det \\Lambda.$$\n\n## Remarks\n\nThe lower bound is elementary: $K$ contains the cross-polytope spanned by $\\pm v_i/\\lambda_i$ for suitable independent lattice vectors $v_i$, and this cross-polytope has volume at least $2^n \\det\\Lambda / (n! \\, \\lambda_1 \\cdots \\lambda_n)$.\nThe upper bound is the deep part; Minkowski's proof compares $K$ with a body built from the successive minima and uses a packing argument.\nSince $\\lambda_1^n \\le \\lambda_1 \\cdots \\lambda_n$, the upper bound implies $\\lambda_1^n \\operatorname{vol}(K) \\le 2^n \\det \\Lambda$; so if $\\operatorname{vol}(K) \\ge 2^n \\det\\Lambda$, then $\\lambda_1 \\le 1$ and $K$ contains a nonzero lattice point, which is [[Minkowski's theorem|0K387S2PKQ]].\nBoth bounds are sharp for $\\Lambda = \\Z^n$: for the cube $K = [-1, 1]^n$ all $\\lambda_k = 1$ and $\\operatorname{vol}(K) = 2^n$, and for the cross-polytope $\\set{x : \\abs{x_1} + \\dots + \\abs{x_n} \\le 1}$ all $\\lambda_k = 1$ and $\\operatorname{vol}(K) = 2^n/n!$.\n\n## Explanations\n\n### L1 human\n\nBlow up a symmetric convex body around the origin until it contains one lattice direction, then two independent ones, and so on.\nThe scaling factors needed are the successive minima.\nThe theorem says that their product, times the volume of the body, is about the volume of a lattice cell, up to factors that depend only on the dimension.\n\n### L2 human\n\nMinkowski's first theorem controls only the first minimum; the second theorem controls all of them at once.\nThis is what one needs to find a whole basis of short vectors.\nIt is used, for instance, in the reduction theory of quadratic forms and in bounds for units and ideals in number fields.\nIt is also the starting point of transference theorems that relate a lattice to its dual.\n",
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