Given a measurable subset , we denote its
-dimensional Lebesgue measure by
. We will denote by
the open ball centered at the origin and having the same measure as
, i.e.
. The norm of vector
will be denoted by
. Finally, we will denote by
the volume of the unit ball in
. It is worth recalling that
where us the usual gamma function.
Definition (Schwarz symmetrization). Let
be a bounded domain. Let
be a measurable function. Then, its Schwarz symmetrization (or the spherically symmetric and decreasing rearrangement) is the function
defined by
.
Observe that if is the radius of
, then
We obviously have the following properties of Schwarz symmetrization (more…)