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  "body": "\n## Statement\n\nLet $d \\ge 1$ and let $X \\subseteq \\R^d$ be a finite set with at least $d + 2$ points.\nThen $X$ can be [[partitioned|08QMKTWF9T]] into two disjoint sets $A$ and $B$ whose [[convex hulls|0FQSDC2EQ2]] intersect:\n$$\\operatorname{conv}(A) \\cap \\operatorname{conv}(B) \\ne \\emptyset.$$\n\n## Remarks\n\nProof idea: take $d + 2$ points $x_1, \\dots, x_{d+2}$ of $X$.\nThe $d + 2$ vectors $(x_i, 1) \\in \\R^{d+1}$ are linearly dependent, so there are reals $\\lambda_i$, not all $0$, with $\\sum \\lambda_i x_i = 0$ and $\\sum \\lambda_i = 0$.\nLet $A$ hold the points with $\\lambda_i \u003e 0$ and let $B$ hold the rest, and put $s = \\sum_{\\lambda_i \u003e 0} \\lambda_i \u003e 0$.\nThen $\\sum_{\\lambda_i \u003e 0} (\\lambda_i / s) x_i = \\sum_{\\lambda_i \\le 0} (-\\lambda_i / s) x_i$ is a point of both hulls; the remaining points of $X$ can be added to either part.\n\nExample: take four points in the plane.\nIf one of them lies in the convex hull of the other three, that point against the other three is a Radon partition.\nOtherwise they are the vertices of a convex quadrilateral, and its two diagonals cross.\n\nThe number $d + 2$ is sharp: the $d + 1$ vertices of a simplex in $\\R^d$ are affinely independent, and no partition of them has intersecting hulls.\n\n## Explanations\n\n### L1 human\n\nTake any $d + 2$ points in $d$-dimensional space, for example four points in the plane.\nYou can always colour them red and blue so that the red shape and the blue shape overlap.\nWith three points in the plane this can fail: the corners of a triangle cannot be split this way.\n\n### L2 human\n\nRadon's theorem is the key step in a short proof of Helly's theorem.\nIt also shows that half-spaces in $\\R^d$ have VC dimension at most $d + 1$ in learning theory, since no set of $d + 2$ points can be shattered.\nIts generalisation to partitions into more than two parts is Tverberg's theorem, and a topological version replaces convex hulls by images of faces of a simplex under continuous maps.\n",
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