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  "body": "\n## Summary\n\nAn $\\varepsilon$-net of a [[family of sets|0C21Q694AG]] is a subset of the ground set that meets every large member of the family.\nHere large means that the member holds at least an $\\varepsilon$ fraction of the points.\nIt is a small sample that is guaranteed to hit every heavy range.\nThe $\\varepsilon$-net theorem of Haussler and Welzl says that families of [[VC dimension|KCFXR9K3GD]] $d$ have $\\varepsilon$-nets of size $O\\big(\\frac{d}{\\varepsilon} \\log \\frac{d}{\\varepsilon}\\big)$, independent of the number of points.\nThis makes $\\varepsilon$-nets a central tool in computational geometry and learning theory.\n\n## Explanations\n\n### L1 human\n\nSuppose there are many points in the plane and many possible regions, such as all half-planes.\nAn $\\varepsilon$-net is a handful of the points chosen so that every region containing a fair share of all points contains at least one chosen point.\nFor example, take $100$ points on a line, with intervals as the ranges.\nThe points in positions $10, 20, \\dots, 100$ form a $1/10$-net: any interval with at least $10$ of the points contains one of these.\nA random sample of the right size works with high probability when the regions are simple.\nThis is unrelated to the $\\varepsilon$-nets of metric spaces, which cover a space by small balls.\n",
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