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.
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
.
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.
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
.
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.
