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  "body": "\n## Statement\n\nLet $n \\ge 1$, and let $S \\subseteq \\R^n$ be a bounded set with [[diameter|0KW3PN547K]] $d$.\nThen $S$ is contained in a [[closed ball|0Q0H13G8ZB]] of radius\n$$R = d \\sqrt{\\frac{n}{2(n+1)}}.$$\n\n## Remarks\n\nTake the smallest closed ball containing $S$; by a compactness argument its centre lies in the convex hull of the points of the closure of $S$ on its boundary sphere.\nSince this centre is a convex combination of at most $n + 1$ such points, a computation with these points and the inequality $\\abs{x_i - x_j} \\le d$ gives the bound.\nA classical alternative uses [[Helly's theorem|F1W3QRBNGJ]] to reduce to sets of $n + 1$ points.\nThe bound is sharp: the vertices of a regular $n$-simplex of edge $d$ have diameter $d$ and circumradius exactly $d \\sqrt{n / (2(n+1))}$.\nExample: for $n = 2$ the radius is $d / \\sqrt{3}$, the circumradius of an equilateral triangle of side $d$.\nFor $n = 1$ the radius is $d/2$, as expected for a segment.\n\n## Explanations\n\n### L1 human\n\nAny set of points whose pairwise distances are at most $d$ fits inside a ball of a fixed radius, a little larger than $d/2$.\nIn the plane the radius $d/\\sqrt{3}$ suffices, and the equilateral triangle shows that nothing smaller always works.\n\n### L2 human\n\nThe trivial bound is $R \\le d$: a ball of radius $d$ centred at any point of $S$ contains $S$.\nJung's theorem gives the optimal constant, which grows from $1/2$ for $n = 1$ towards $1/\\sqrt{2}$ as $n \\to \\infty$.\nIt is a typical result of convex geometry that relates two measures of size, the diameter and the circumradius.\nIt is often used together with the isodiametric inequality and in Borsuk-type partition problems.\n",
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