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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $A \\subseteq \\R^n$ be nonempty.\nThe *support function* of $A$ is the map $h_A \\colon \\R^n \\to \\R \\cup \\set{+\\infty}$ given by\n$$h_A(x) = \\sup_{a \\in A} \\, x \\cdot a,$$\nwhere $x \\cdot a$ is the [[dot product|0S52TNNTH9]] and $\\sup$ is the [[supremum|0R3H5PANRQ]].\n\n## Remarks\n\nIf $A$ is bounded, then $h_A$ takes only real values.\n\nExample: for $A = [0, 1]^2$, $h_A(x_1, x_2) = \\max(x_1, 0) + \\max(x_2, 0)$.\nIndeed $x \\cdot a = x_1 a_1 + x_2 a_2$, and each term is largest at $a_i = 1$ when $x_i \u003e 0$ and at $a_i = 0$ when $x_i \\le 0$.\n\nNon-example: the function $f(x) = \\abs{x}^2$ is not the support function of any nonempty set.\nEvery support function satisfies $h_A(t x) = t\\, h_A(x)$ for $t \u003e 0$, but $f(2x) = 4 f(x) \\ne 2 f(x)$ for $x \\ne 0$.\n\nEach support function is positively homogeneous and subadditive, $h_A(x + y) \\le h_A(x) + h_A(y)$, hence convex.\nOne has $h_A = h_B$ when $B$ is the closure of the convex hull of $A$, and a nonempty compact convex set is determined by its support function.\n",
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