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  "body": "\n## Statement\n\nLet $d \\ge 1$ and let $C_1, \\dots, C_m$ be [[convex sets|02AS45T3H9]] in $\\R^d$, with $m \\ge d + 1$.\nSuppose that every $d + 1$ of the sets have a point in common.\nThen all $m$ sets have a point in common: $C_1 \\cap \\cdots \\cap C_m \\ne \\emptyset$.\n\n## Remarks\n\nProof idea: induct on $m$, starting from $m = d + 1$.\nFor $m \\ge d + 2$, the induction hypothesis gives, for each $j$, a point $x_j$ in all the sets except possibly $C_j$.\n[[Radon's theorem|888R24ZGNY]] applied to $x_1, \\dots, x_m$ gives a partition into $A$ and $B$ with a common point $p$ of their convex hulls.\nEach $C_i$ contains all of $A$ or all of $B$, so it contains $p$ by convexity.\n\nExample: if every three of finitely many convex sets in the plane meet, then all of them meet.\n\nThe number $d + 1$ cannot be lowered: the three lines that contain the sides of a triangle in $\\R^2$ meet in pairs but have no common point.\nFor an infinite family the statement can fail: the half-lines $[k, \\infty)$, $k = 1, 2, \\dots$, in $\\R$ meet in pairs but have empty intersection.\nThe infinite version holds when all the sets are [[compact|0R6EWFKRCF]] and convex.\n\n## Explanations\n\n### L1 human\n\nTo check that many convex sets share a point, it is enough to check small groups.\nIn the plane, groups of three suffice; in space, groups of four.\nWithout convexity this fails.\nCut a circle in the plane into four disjoint arcs, and for each arc let $S_i$ be the circle with that arc removed.\nAny three of the sets $S_i$ share the fourth arc, but no point lies in all four.\n\n### L2 human\n\nHelly's theorem is the model of a Helly-type theorem: a global property follows from the same property for all small subfamilies.\nIt has many consequences, such as the centerpoint theorem and Jung's theorem on the smallest ball containing a set.\nIt is used in Kirchberger's theorem and has a fractional version and a colourful version.\nIn optimisation it explains why an infeasible system of convex constraints in $\\R^d$ has an infeasible subsystem with at most $d + 1$ constraints.\n",
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