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  "body": "\n## Summary\n\nThe successive minima of a [[lattice|WN9V7WSJKF]] with respect to a symmetric convex body are $n$ numbers $\\lambda_1 \\le \\dots \\le \\lambda_n$.\nThe $k$-th one is the smallest scaling factor at which the scaled body contains $k$ linearly independent lattice vectors.\nThey measure how long the short vectors of the lattice are, in all independent directions at once.\n[[Minkowski's second theorem|5H4XZVF9B9]] bounds their product in terms of the volume of the body and the covolume of the lattice.\n\n## Explanations\n\n### L1 human\n\nTake a grid of points and a symmetric convex shape centred at the origin, such as a disc or a box.\nInflate the shape slowly.\nAt some moment it first catches a grid point other than the origin; the scaling factor at that moment is $\\lambda_1$.\nKeep inflating until it holds two grid points that do not lie on one line through the origin; that factor is $\\lambda_2$, and so on.\nFor the integer grid $\\Z^2$ and the box $[-1, 1] \\times [-2, 2]$, the point $(0, 1)$ enters at factor $1/2$ and the point $(1, 0)$ at factor $1$, so $\\lambda_1 = 1/2$ and $\\lambda_2 = 1$.\n",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/YDM/YDMBKD8FP2.md",
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