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  "body": "\n## Summary\n\nA supporting hyperplane of a set $S$ in $\\R^n$ is a hyperplane that has all of $S$ on one side.\nIt also touches $S$ at a point of the [[boundary|05D5EGMFYV]] of $S$.\nBy the [[supporting hyperplane theorem|DR9BXFWMQB]], a [[convex set|02AS45T3H9]] has a supporting hyperplane through each of its boundary points.\nSupporting hyperplanes link convex sets with linear functions: they describe where a linear function is largest on the set.\n\n## Explanations\n\n### L1 human\n\nPut a convex object on a table.\nThe table top is a supporting plane: the object lies entirely above it and touches it.\nIn the plane, a supporting line of a disc is a tangent line, and a supporting line of a square through a corner can be tilted in many ways.\nFor the square $[0, 1]^2$, the line $x = 1$ is a supporting line touching the right side.\n",
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