In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the symmetric decreasing rearrangement of a function is a function which is symmetric and decreasing, and whose
level sets
In mathematics, a level set of a real-valued function of real variables is a set where the function takes on a given constant value , that is:
: L_c(f) = \left\~,
When the number of independent variables is two, a level set is calle ...
are of the same size as those of the original function.
Definition for sets
Given a
measurable set
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simila ...
,
in
one defines the ''symmetric rearrangement'' of
called
as the ball centered at the origin, whose volume (
Lebesgue measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...
) is the same as that of the set
An equivalent definition is
where
is the volume of the
unit ball
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* ''Unit'' (alb ...
and where
is the volume of
Definition for functions
The rearrangement of a non-negative, measurable real-valued function
whose level sets
(for
) have finite measure is
where
denotes the
indicator function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\i ...
of the set
In words, the value of
gives the height
for which the radius of the symmetric
rearrangement of
is equal to
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
The function
is a symmetric and decreasing function whose level sets have the same measure as the level sets of
that is,
If
is a function in
then
The
Hardy–Littlewood inequality In mathematical analysis, the Hardy–Littlewood inequality, named after G. H. Hardy and John Edensor Littlewood, states that if f and g are nonnegative measurable real functions vanishing at infinity that are defined on n-dimensional Euclidean spa ...
holds, that is,
Further, the
Pólya–Szegő inequality holds. This says that if
and if
then
The symmetric decreasing rearrangement is order preserving and decreases
distance, that is,
and
Applications
The Pólya–Szegő inequality yields, in the limit case, with
the
isoperimetric inequality
In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n ...
. Also, one can use some relations with harmonic functions to prove the
Rayleigh–Faber–Krahn inequality In spectral geometry, the Rayleigh–Faber–Krahn inequality, named after its conjecturer, Lord Rayleigh, and two individuals who independently proved the conjecture, G. Faber and Edgar Krahn, is an inequality concerning the lowest Dirichlet eigen ...
.
Nonsymmetric decreasing rearrangement
We can also define
as a function on the nonnegative real numbers rather than on all of
Let
be a
σ-finite measure space, and let