# Symmetric decreasing rearrangement

In mathematics, the symmetric decreasing rearrangement of a function is a function which is symmetric and decreasing, and whose level sets are of the same size as those of the original function.[1]

## Definition for sets

Given a measurable set, ${\displaystyle A}$, in Rn one can obtain the symmetric rearrangement of ${\displaystyle A}$, called ${\displaystyle A^{*}}$, by

${\displaystyle A^{*}=\{x\in \mathbf {R} ^{n}:\,\omega _{n}\cdot |x|^{n}<|A|\},}$

where ${\displaystyle \omega _{n}}$ is the volume of the unit ball and where ${\displaystyle |A|}$ is the volume of ${\displaystyle A}$. Notice that this is just the ball centered at the origin whose volume is the same as that of the set ${\displaystyle A}$.

## Definition for functions

The rearrangement of a non-negative, measurable real-valued function ${\displaystyle f}$ whose level sets ${\displaystyle f^{-1}(y)}$ (${\displaystyle y\in \mathbb {R} _{\geq 0}}$) have finite measure is

${\displaystyle f^{*}(x)=\int _{0}^{\infty }\mathbb {I} _{\{y:f(y)>t\}^{*}}(x)\,dt,}$

where ${\displaystyle \mathbb {I} _{A}}$ denotes the indicator function of the set A. In words, the value of ${\displaystyle f^{*}(x)}$ gives the height t for which the radius of the symmetric rearrangement of ${\displaystyle \{y:f(y)>t\}}$ is equal to x. We have the following motivation for this definition. Because the identity

${\displaystyle g(x)=\int _{0}^{\infty }\mathbb {I} _{\{y:g(y)>t\}}(x)\,dt,}$

holds for any non-negative function ${\displaystyle g}$, the above definition is the unique definition that forces the identity ${\displaystyle \mathbb {I} _{A}^{*}=\mathbb {I} _{A^{*}}}$ to hold.

## Properties

The function ${\displaystyle f^{*}}$ is a symmetric and decreasing function whose level sets have the same measure as the level sets of ${\displaystyle f}$, i.e.

${\displaystyle |\{x:f^{*}(x)>t\}|=|\{x:f(x)>t\}|.}$

If ${\displaystyle f}$ is a function in ${\displaystyle L^{p}}$, then

${\displaystyle \|f\|_{L^{p}}=\|f^{*}\|_{L^{p}}.}$

The Hardy–Littlewood inequality holds, i.e.

${\displaystyle \int fg\leq \int f^{*}g^{*}.}$

Further, the Szegő inequality holds. This says that if ${\displaystyle 1\leq p<\infty }$ and if ${\displaystyle f\in W^{1,p}}$ then

${\displaystyle \|\nabla f^{*}\|_{p}\leq \|\nabla f\|_{p}.}$

The symmetric decreasing rearrangement is order preserving and decreases ${\displaystyle L^{p}}$ distance, i.e.

${\displaystyle f\leq g\Rightarrow f^{*}\leq g^{*}}$

and

${\displaystyle \|f-g\|_{L^{p}}\geq \|f^{*}-g^{*}\|_{L^{p}}.}$

## Applications

The Pólya–Szegő inequality yields, in the limit case, with ${\displaystyle p=1}$, the isoperimetric inequality. Also, one can use some relations with harmonic functions to prove the Rayleigh–Faber–Krahn inequality.