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.
