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, $A$ , in Rⁿ one can obtain the symmetric rearrangement of $A$ , called $A^*$ , by

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

where $\omega_n$ 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 function $f$ whose level sets have finite measure is

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

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

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

\int fg \leq \int f^* g^* .

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

\|\nabla f^*\|_p \leq \|\nabla f\|_p.

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

f \leq g \Rightarrow f^* \leq g^*

and

\|f - g\|_{L^p} \geq \|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