# Symmetric decreasing rearrangement

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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.

## Definition for sets

Given a measurable set, A , in Rn one can obtain the symmetric rearrangement of A , called A , by

A = { x R n : ω n | x | n < | A | } ,

where ω n is the volume of the unit ball and where | A | is the volume of A . Notice that this is just the ball centered at the origin whose volume is the same as that of the set A .

## Definition for functions

The rearrangement of a non-negative, measurable real-valued function f whose level sets f 1 ( y ) ( y R 0 ) have finite measure is

f ( x ) = 0 I { y : f ( y ) > t } ( x ) d t ,

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

g ( x ) = 0 I { y : g ( y ) > t } ( x ) d t ,

holds for any non-negative function g , the above definition is the unique definition that forces the identity I A = I A to hold.

## Properties

The function f is a symmetric and decreasing function whose level sets have the same measure as the level sets of f , i.e.

| { x : f ( x ) > t } | = | { x : f ( x ) > t } | .

If f is a function in L p , then

f L p = f L p .

The Hardy–Littlewood inequality holds, i.e.

f g f g .

Further, the Szegő inequality holds. This says that if 1 p < and if f W 1 , p then

f p f p .

The symmetric decreasing rearrangement is order preserving and decreases L p distance, i.e.

f g f g

and

f g L p f g L p .

## Applications

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

## References

Symmetric decreasing rearrangement Wikipedia

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