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  "body": "\n## Summary\n\nThe support function of a nonempty set $A \\subseteq \\R^n$ assigns to each vector $x$ the [[supremum|0R3H5PANRQ]] of the [[dot products|0S52TNNTH9]] $x \\cdot a$ over $a \\in A$.\nIt is a [[sublinear function|K3356N0SAX]], and it determines a nonempty compact [[convex set|02AS45T3H9]] completely.\nSupport functions turn geometric operations into analytic ones: the support function of a Minkowski sum is the sum of the support functions.\nThey are a basic tool in the study of [[convex bodies|437FSG3419]].\n\n## Explanations\n\n### L1 human\n\nFix a direction $u$ of length $1$ and push a flat wall, perpendicular to $u$, towards the set from far away in direction $u$.\nThe value of the support function at $u$ is the signed distance from the origin to the wall at the moment it first touches the set.\nFor the unit disc centred at the origin, the wall always touches at distance $1$, so the support function is $h(u) = 1$ on unit vectors, and $h(x) = \\abs{x}$ in general.\nFor the segment from $(-1, 0)$ to $(1, 0)$, one gets $h(x_1, x_2) = \\abs{x_1}$.\n",
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