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  "body": "\n## Definition\n\nLet $\\abs{\\cdot}$ denote [[Lebesgue measure|2Z2QVWZ262]] on $\\R^n$.\nFor a measurable set $A \\subseteq \\R^n$ of finite measure, its *symmetric rearrangement* $A^*$ is the open ball centred at $0$ with $\\abs{A^*} = \\abs{A}$, and $A^* = \\varnothing$ if $\\abs{A} = 0$.\nLet $f \\colon \\R^n \\to \\C$ be a [[measurable function|YYNCJSR0PJ]] that vanishes at infinity, that is, $\\abs{\\set{x : \\abs{f(x)} \u003e t}} \u003c \\infty$ for every $t \u003e 0$.\nThe *symmetric decreasing rearrangement* of $f$ is the function $f^* \\colon \\R^n \\to [0, \\infty]$ given by\n$$f^*(x) = \\int_0^\\infty \\mathbf{1}_{\\set{\\abs{f} \u003e t}^*}(x) \\, dt,$$\nwhere $\\mathbf{1}_B$ is the indicator function of $B$.\n\n## Remarks\n\nThe function $f^*$ is nonnegative, radial, non-increasing in $\\abs{x}$ and lower semicontinuous.\nFor every $t \u003e 0$, $\\set{f^* \u003e t} = \\set{\\abs{f} \u003e t}^*$, so $f^*$ and $\\abs{f}$ have the same distribution function, and $\\norm{f^*}_p = \\norm{f}_p$ for $1 \\le p \\le \\infty$.\nThe Pólya–Szegő inequality states that $\\norm{\\nabla f^*}_2 \\le \\norm{\\nabla f}_2$ for $f \\in H^1(\\R^n)$.\nThe Riesz rearrangement inequality states that $\\iint f(x) g(x - y) h(y) \\, dx \\, dy \\le \\iint f^*(x) g^*(x - y) h^*(y) \\, dx \\, dy$ for nonnegative $f, g, h$.\n\nExample: in $\\R$, let $f = \\mathbf{1}_{[1, 3]}$.\nFor $0 \u003c t \u003c 1$ the set $\\set{f \u003e t}$ is $[1, 3]$, of length $2$, so its rearrangement is $(-1, 1)$; for $t \\ge 1$ it is empty.\nHence $f^* = \\mathbf{1}_{(-1, 1)}$.\n\nNon-example: the constant function $f = 1$ on $\\R^n$ does not vanish at infinity, since $\\set{f \u003e 1/2}$ has infinite measure, so $f^*$ is not defined.\n\nThe one-variable [[decreasing rearrangement|BWS042NC2P]] $f^\\sharp$ on $[0, \\infty)$ is related by $f^*(x) = f^\\sharp(\\omega_n \\abs{x}^n)$, where $\\omega_n$ is the volume of the unit ball, up to values on a null set.\n",
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