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.[1]

Definition for sets[edit]

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

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

Definition for functions[edit]

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

where denotes the indicator function of the set A. In words, the value of gives the height t for which the radius of the symmetric rearrangement of is equal to x. We have the following motivation for this definition. Because the identity

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

Properties[edit]

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

If is a function in , then

The Hardy–Littlewood inequality holds, i.e.

Further, the Szegő inequality holds. This says that if and if then

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

and

Applications[edit]

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

See also[edit]

References[edit]

  1. ^ Lieb, Elliott; Loss, Michael (2001). Analysis. Graduate Studies in Mathematics. 14 (2nd ed.). American Mathematical Society. ISBN 978-0821827833.