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  "body": "\n## Statement\n\nLet $(S, \\Sigma, \\mu)$ be a [[measure space|P15VQS5DJ9]] and let $p \\in [1, \\infty)$.\nLet $f, g \\colon S \\to \\R$ be [[measurable functions|YYNCJSR0PJ]] with $\\int_S \\abs{f}^p \\, d\\mu \u003c \\infty$ and $\\int_S \\abs{g}^p \\, d\\mu \u003c \\infty$.\nThen\n$$\\Big( \\int_S \\abs{f + g}^p \\, d\\mu \\Big)^{1/p} \\le \\Big( \\int_S \\abs{f}^p \\, d\\mu \\Big)^{1/p} + \\Big( \\int_S \\abs{g}^p \\, d\\mu \\Big)^{1/p}.$$\n\n## Remarks\n\nProof idea: the case $p = 1$ follows from $\\abs{f + g} \\le \\abs{f} + \\abs{g}$.\nFor $p \u003e 1$, first $\\abs{f + g}^p \\le 2^{p-1}(\\abs{f}^p + \\abs{g}^p)$ shows that the left side is finite.\nThen write $\\abs{f + g}^p \\le \\abs{f} \\abs{f + g}^{p-1} + \\abs{g} \\abs{f + g}^{p-1}$ and apply [[Hölder's inequality|JQ9WXC729D]] to each term with $q = p / (p - 1)$.\n\nWith the counting measure on $\\set{1, \\dots, n}$ it shows that the [[p-norm on R^n|0KNJWEVSMG]] satisfies the triangle inequality.\nIn general it is the triangle inequality for the [[norm|08NYZ2FS6B]] of the [[Lp space|SASP14MA1Z]] $L^p(\\mu)$.\n\nThe hypothesis $p \\ge 1$ is needed.\nFor $p = 1/2$ on $\\R^2$ with the counting measure, take $f = (1, 0)$ and $g = (0, 1)$: the left side is $(1 + 1)^2 = 4$ and the right side is $1 + 1 = 2$.\n\n## Explanations\n\n### L1 human\n\nThe Minkowski inequality is the triangle inequality for the $p$-size of functions or vectors.\nThe size of a sum is at most the sum of the sizes.\nFor $p = 2$ and vectors in the plane, it says that one side of a triangle is at most the sum of the other two.\n\n### L2 human\n\nThis inequality is what makes $\\norm{f}_p = (\\int \\abs{f}^p \\, d\\mu)^{1/p}$ a norm for $p \\ge 1$, so that $L^p$ is a normed space.\nIts proof is a short application of Hölder's inequality.\nFor $0 \u003c p \u003c 1$ the inequality fails, and the quantity $\\norm{f}_p$ is then not a norm.\n",
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