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  "body": "\n## Statement\n\nLet $d \\ge 1$ and let $X$ be a finite set of $n$ points in $\\R^d$.\nThen there is a point $c \\in \\R^d$ such that every closed half-space of $\\R^d$ that contains $c$ contains at least $\\dfrac{n}{d+1}$ points of $X$.\nSuch a point $c$ is called a *centerpoint* of $X$.\n\n## Remarks\n\nConsider all closed half-spaces that contain more than $\\frac{d}{d+1} n$ points of $X$, and the [[convex hulls|0FQSDC2EQ2]] of the points of $X$ in each of them.\nEach such half-space misses fewer than $\\frac{n}{d+1}$ points of $X$, so $d + 1$ of them together miss fewer than $n$ points, and some point of $X$ lies in all $d + 1$ hulls.\nBy [[Helly's theorem|F1W3QRBNGJ]] all these hulls have a common point $c$.\nSuppose a closed half-space containing $c$ had fewer than $\\frac{n}{d+1}$ points.\nIts open complement would contain more than $\\frac{d}{d+1} n$ points, and a slightly smaller closed half-space inside it would give a hull that misses $c$, a contradiction.\nIn dimension $1$ a median of $X$ is a centerpoint, with the bound $n/2$.\nThe bound $n/(d+1)$ cannot be improved in general.\nTake $d + 1$ equal small clusters of points near the vertices of a simplex: for any point $c$, some closed half-space through $c$ meets only one cluster.\n\n## Explanations\n\n### L1 human\n\nA centerpoint is a point that sits \"deep\" inside a cloud of points.\nHowever one cuts space in two halves by a flat wall through it, each half keeps a fair share of the points.\nIn the plane that fair share is at least one third of the points.\nOn a line it is the median, which has half the points on each side.\n\n### L2 human\n\nThe centerpoint theorem is a standard consequence of [[Helly's theorem|F1W3QRBNGJ]].\nIt generalizes the median to higher dimensions and gives a notion of depth for data.\nIt is used in divide-and-conquer algorithms and in the proof of separator theorems for planar graphs and meshes.\nThe [[ham sandwich theorem|GDXX60SNG5]] is a different statement about dividing point sets by hyperplanes.\n",
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