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  "body": "\n## Definition\n\nLet $n \\ge 1$.\nFor $x \\in \\R^n$, let $x_{[1]} \\ge x_{[2]} \\ge \\dots \\ge x_{[n]}$ be the entries of $x$ in decreasing order.\nA vector $y \\in \\R^n$ *majorizes* $x \\in \\R^n$, written $x \\prec y$, if\n$$\\sum_{i=1}^{k} x_{[i]} \\le \\sum_{i=1}^{k} y_{[i]} \\quad \\text{for } k = 1, \\dots, n - 1, \\qquad \\sum_{i=1}^{n} x_{[i]} = \\sum_{i=1}^{n} y_{[i]};$$\nsee [[majorization|5K7VT2VKZT]].\nA function $f \\colon \\R^n \\to \\R$ is *Schur-convex* if $f(x) \\le f(y)$ for all $x, y \\in \\R^n$ with $x \\prec y$.\nIt is *Schur-concave* if $-f$ is Schur-convex.\n\n## Remarks\n\nA Schur-convex function is symmetric: if $y$ is obtained from $x$ by a [[permutation|0BQMWCNS8S]] of the entries, then $x \\prec y$ and $y \\prec x$, so $f(x) = f(y)$.\nEvery symmetric convex function is Schur-convex.\nFor a symmetric $f$ with continuous partial derivatives on $\\R^n$, the Schur–Ostrowski criterion says that $f$ is Schur-convex if and only if $(x_i - x_j)\\big(\\partial_i f(x) - \\partial_j f(x)\\big) \\ge 0$ for all $x$ and all $i, j$.\n\nExample: $f(x) = \\max_i x_i = x_{[1]}$.\nIf $x \\prec y$, the case $k = 1$ of the definition gives $x_{[1]} \\le y_{[1]}$, so $f(x) \\le f(y)$.\n\nNon-example: $f(x_1, x_2) = x_1$ on $\\R^2$.\nThe vectors $(1, 0)$ and $(0, 1)$ majorize each other, but $f(1, 0) = 1 \u003e 0 = f(0, 1)$.\n",
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