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  "body": "\n## Statement\n\nLet $P$ be a subset of $\\mathbb{R}^d$, and let $x$ be a point of its [[convex hull|0FQSDC2EQ2]] $\\operatorname{conv}(P)$.\nThen there are at most $d + 1$ points $p_1, \\dots, p_k$ of $P$, with $k \\leq d + 1$, such that $x \\in \\operatorname{conv}\\{p_1, \\dots, p_k\\}$.\nEquivalently, $x$ is a convex combination of at most $d + 1$ points of $P$.\n\n## Remarks\n\nThus $\\operatorname{conv}(P)$ is the union of the simplices with vertices in $P$, of dimension at most $d$. Mathlib states this form as `convexHull_eq_union`.\nThe bound $d + 1$ is the best possible in general. Let $P$ be the three vertices of a triangle in $\\mathbb{R}^2$. A point inside the triangle is not in the hull of two points of $P$.\nThe proof only uses the order and the field operations of $\\mathbb{R}$. So it works over each ordered field.\n\n## History\n\nCarathéodory proved the theorem in 1911 for compact sets $P$. Steinitz extended it to arbitrary sets in 1914.\n\n## Explanations\n\n### L1 human\n\nIn the plane, a point in the convex hull of many points is always inside a triangle with corners from those points. You never need more than three corners. In space, four corners, the vertices of a tetrahedron, are always enough.\n",
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