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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.<ref name=liebloss>{{cite book | author=Lieb, Elliott H., &amp; Loss, Michael | title=Analysis | edition=Second | publisher=American Mathematical Society | location=Providence, RI | year=2001 | isbn=0-8218-2783-9 }}</ref>

== Definition for sets ==
Given a [[measurable set]], <math>A</math>, in '''R'''<sup>''n''</sup> one can obtain the ''symmetric rearrangement'' of <math> A </math>, called <math>A^*</math>, by

:<math> A^* = \{x \in \mathbf{R}^n :\,\omega_n\cdot|x|^n < |A| \},</math>

where <math>\omega_n</math> is the volume of the [[unit ball]] and where <math>|A|</math> is the volume of <math>A</math>.  Notice that this is just the ball centered at the origin whose volume is the same as that of the set <math> A </math>.

== Definition for functions ==
The rearrangement of a non-negative, measurable function <math>f</math> whose level sets have finite measure is

:<math> f^*(x) = \int_0^\infty \mathbb{I}_{\{y: f(y)>t\}^*}(x) \, dt.</math>

We have the following motivation for this definition.  Because the identity

:<math> g(x) = \int_0^\infty \mathbb{I}_{\{y: g(y)>t\}}(x) \, dt,</math>

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

== Properties ==
The function <math>f^*</math> is a symmetric and decreasing function whose level sets have the same measure as the level sets of <math> f</math>, i.e.

:<math> |\{ x: f^*(x)>t\}| = |\{x: f(x)>t\}|.</math>

If <math>f</math> is a function in <math> L^p</math>, then

:<math> \|f\|_{L^p} = \|f^*\|_{L^p}.</math>

The [[Hardy–Littlewood inequality]] holds, i.e.

:<math> \int fg \leq \int f^* g^* .</math>

Further, the [[Szegő inequality]] holds.  This says that if <math>1 \leq p < \infty </math> and if <math> f\in W^{1,p} </math> then

:<math> \|\nabla f^*\|_p \leq \|\nabla f\|_p.</math>

The symmetric decreasing rearrangement is order preserving and decreases <math> L^p</math> distance, i.e.

:<math> f \leq g \Rightarrow  f^* \leq g^* </math>

and

:<math> \|f - g\|_{L^p} \geq \|f^* - g^*\|_{L^p}.</math>

== Applications ==
The Pólya&ndash;Szegő inequality yields, in the limit case, with <math> p = 1</math>, the [[isoperimetric inequality]].  Also, one can use some relations with harmonic functions to prove the [[Rayleigh–Faber–Krahn inequality]].

== See also ==
*[[Isoperimetric inequality]]
*[[Layer cake representation]]
*[[Rayleigh–Faber–Krahn inequality]]
*[[Riesz rearrangement inequality]]
*[[Sobolev space]]
*[[Szegő inequality]]

==References==
{{reflist}}

[[Category:Multivariable calculus]]
[[Category:Real analysis]]