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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $\\Lambda \\subset \\R^n$ be a [[lattice|WN9V7WSJKF]].\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 $\\lambda \u003e 0$ write $\\lambda K = \\set{\\lambda x : x \\in K}$.\nLet $k \\in \\set{1, \\dots, n}$.\nThe *$k$-th successive minimum* $\\lambda_k(K, \\Lambda)$ of $\\Lambda$ for $K$ is the infimum of all $\\lambda \u003e 0$ such that $\\lambda K \\cap \\Lambda$ contains $k$ [[linearly independent|0CRMGPAH83]] vectors.\n\n## Remarks\n\nThe infimum is attained, and $0 \u003c \\lambda_1 \\le \\lambda_2 \\le \\dots \\le \\lambda_n \u003c \\infty$.\nExample: for $\\Lambda = \\Z^2$ and $K = [-1, 1] \\times [-2, 2]$, one has $\\lambda K = [-\\lambda, \\lambda] \\times [-2\\lambda, 2\\lambda]$.\nThe vector $(0, 1)$ lies in $\\lambda K$ exactly when $\\lambda \\ge 1/2$, and no nonzero integer vector lies in $\\lambda K$ for $\\lambda \u003c 1/2$, so $\\lambda_1 = 1/2$.\nA vector independent of $(0, 1)$ has first coordinate of absolute value at least $1$, so it needs $\\lambda \\ge 1$, and $(1, 0)$ enters at $\\lambda = 1$; so $\\lambda_2 = 1$.\nNon-example: the strip $K = \\R \\times [-1, 1]$ is convex and symmetric but not compact.\nThen $(1, 0) \\in \\lambda K$ for every $\\lambda \u003e 0$, so the infimum defining $\\lambda_1$ is $0$ and the minima lose their meaning.\nWhen $K$ is the closed unit ball, $\\lambda_k$ is the smallest $r$ such that $\\Lambda$ has $k$ linearly independent vectors of length at most $r$.\n",
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