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  "body": "\n## Definition\n\nLet $X$ be a subset of $\\mathbb{R}^n$. The convex hull of $X$ is the intersection of all [[convex sets|02AS45T3H9]] of $\\mathbb{R}^n$ that contain $X$:\n$$\\operatorname{conv}(X) = \\bigcap \\{\\, S \\subseteq \\mathbb{R}^n \\;:\\; S \\supseteq X \\text{ and } S \\text{ is convex} \\,\\}.$$\n\n## Remarks\n\n$\\operatorname{conv}(X)$ is convex, and it is the smallest convex set that contains $X$.\nIt is also the set of all convex combinations $\\sum_{i=1}^{k} w_i x_i$, with $k \\geq 1$, $x_i \\in X$, $w_i \\geq 0$ and $\\sum_i w_i = 1$. These are the [[affine combinations|0KJH6SRFZK]] with nonnegative weights.\nThe convex hull of a compact set is compact, and the convex hull of an open set is open.\nThe convex hull of a finite set is a convex polytope. For three points not on a line, it is the triangle with its inside.\n\n## Explanations\n\n### L1 human\n\nFor a finite set of points in the plane, put a rubber band round all the points and let it go. The band takes the shape of the boundary of the convex hull. The convex hull of the four corners of a square is the full square.\n",
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