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  "body": "\n## Definition\n\nLet $X$ be a [[vector space|000AYPA9AJ]] over $\\R$.\nA *sublinear function* on $X$ is a map $p \\colon X \\to \\R$ such that:\n\n1. $p(tx) = t\\, p(x)$ for all real $t \u003e 0$ and all $x \\in X$ (positive homogeneity).\n2. $p(x + y) \\le p(x) + p(y)$ for all $x, y \\in X$ (subadditivity).\n\n## Remarks\n\nEvery norm on $X$ is sublinear.\nFor example, $p(x) = \\abs{x}$ on $\\R$ satisfies $\\abs{tx} = t\\abs{x}$ for $t \u003e 0$ and $\\abs{x + y} \\le \\abs{x} + \\abs{y}$.\nEvery linear functional is sublinear, and so is $p(x_1, x_2) = \\max(x_1, x_2)$ on $\\R^2$.\nThe map $p(x) = x^2$ on $\\R$ is not sublinear: $p(2 \\cdot 1) = 4$ but $2\\, p(1) = 2$, so clause 1 fails.\nClause 1 with $t = 2$ and $x = 0$ gives $p(0) = 2p(0)$, so $p(0) = 0$; hence clause 1 also holds for $t = 0$.\nA sublinear function $p$ with $p(-x) = p(x)$ for all $x$ is exactly a [[seminorm|FP5GVCN8GV]].\n",
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