  The paper has two main goals. The first is to take a new approach to
rearrangements on certain classes of measurable real-valued functions on
$\mathbb{R}^n$. Rearrangements are maps that are monotonic (up to sets of
measure zero) and equimeasurable, i.e., they preserve the measure of
super-level sets of functions. All the principal known symmetrization processes
for functions, such as Steiner and Schwarz symmetrization, are rearrangements,
and these have a multitude of applications in diverse areas of the mathematical
sciences. The second goal is to understand which properties of rearrangements
characterize polarization, a special rearrangement that has proved particularly
useful in a number of contexts. In order to achieve this, new results are
obtained on the structure of measure-preserving maps on convex bodies and of
rearrangements generally.
