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  "body": "\n## Statement\n\nLet $d \\ge 1$ and let $P, Q \\subseteq \\R^d$ be finite sets.\nCall $P' \\subseteq P$ and $Q' \\subseteq Q$ strictly separated if the following holds.\nThere are $a \\in \\R^d$ and $b \\in \\R$ with $a \\cdot p \u003c b$ for all $p \\in P'$ and $a \\cdot q \u003e b$ for all $q \\in Q'$, where $a \\cdot x$ is the [[dot product|0S52TNNTH9]].\nSuppose that for every set $T \\subseteq P \\cup Q$ with at most $d + 2$ points, the sets $T \\cap P$ and $T \\cap Q$ are strictly separated.\nThen $P$ and $Q$ are strictly separated.\n\n## Remarks\n\nProof idea: for each $p \\in P$, the set of pairs $(a, b) \\in \\R^{d+1}$ with $a \\cdot p \u003c b$ is an open half-space, hence [[convex|02AS45T3H9]], and likewise for each $q \\in Q$ with $a \\cdot q \u003e b$.\nThe hypothesis says that every $d + 2$ of these half-spaces in $\\R^{d+1}$ have a common point.\nBy [[Helly's theorem|F1W3QRBNGJ]] in $\\R^{d+1}$, all of them have a common point $(a, b)$, which separates $P$ from $Q$.\n\nExample: take finite sets of red and blue points in the plane.\nIf every four of the points can be split by a line according to their colours, then so can all of them.\n\nThe number $d + 2$ cannot be lowered.\nIn the plane, take the red points $(0, 0)$, $(1, 1)$ and the blue points $(1, 0)$, $(0, 1)$.\nNo line separates them, since the two diagonals cross, while every three of the four points can be separated.\n\n## Explanations\n\n### L1 human\n\nYou have red and blue points and want a straight wall with all red points on one side and all blue points on the other.\nKirchberger's theorem says you only need to test small groups: in the plane, groups of four points.\nIf every small group can be split correctly, the whole set can.\n\n### L2 human\n\nThe theorem is a Helly-type result for linear separability.\nIn machine learning it bounds the size of a certificate that two finite classes are not linearly separable: some $d + 2$ points already show it.\nThe proof shows a general pattern: a separation problem becomes a common-point problem for convex sets in a space of parameters, where Helly's theorem applies.\n",
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