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  "body": "\n## Statement\n\nLet $d \\ge 1$, let $S \\subseteq \\R^d$, and let $x$ be a point in the [[interior|0RZH6W98HK]] of the [[convex hull|0FQSDC2EQ2]] of $S$.\nThen there is a set $T \\subseteq S$ with at most $2d$ points such that $x$ lies in the interior of $\\operatorname{conv}(T)$.\n\n## Remarks\n\nIt is an interior analogue of [[Carathéodory's theorem|0201Q4P2B5]], where $d + 1$ points suffice for membership in the hull but $2d$ points are needed for the interior.\n\nThe bound $2d$ is sharp.\nLet $S = \\set{\\pm e_1, \\dots, \\pm e_d}$ be the vertices of the cross-polytope and let $x = 0$.\nIf $T$ omits, say, $e_i$, then every point of $\\operatorname{conv}(T)$ has $i$-th coordinate at most $0$, so $0$ is not interior to $\\operatorname{conv}(T)$.\n\nExample: in the plane, a point inside the convex hull of a finite set lies inside some quadrilateral, or some triangle, with vertices in the set.\n\nThe interior is taken in $\\R^d$, so the hypothesis already forces $\\operatorname{conv}(S)$ to have dimension $d$.\n\n## Explanations\n\n### L1 human\n\nIf a point sits strictly inside the convex hull of many points, a few of them already surround it.\nIn the plane four points always suffice, and in $d$ dimensions $2d$ points suffice.\nThe four points $(\\pm 1, 0)$, $(0, \\pm 1)$ around the origin show that three would not be enough.\n\n### L2 human\n\nThe theorem was proved by Ernst Steinitz, who also extended Carathéodory's theorem from compact to arbitrary sets.\nIt is a standard companion of Carathéodory's and Helly's theorems in combinatorial convexity.\nIt is used for positively spanning sets.\nA finite set positively spans $\\R^d$ exactly when $0$ is in the interior of its convex hull.\nBy the theorem, a subset of at most $2d$ of its vectors then already positively spans $\\R^d$.\n",
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