Symmetric decreasing rearrangement

Summary

The symmetric decreasing rearrangement of a function $f$ on $\R^n$ is a radial function $f^*$, decreasing in $\abs{x}$. Its superlevel sets are balls centred at the origin with the same volumes as the superlevel sets of $\abs{f}$. It keeps every $L^p$ norm, does not increase the Dirichlet energy, and does not decrease certain double integrals. So it is the main tool to show that balls and radial functions are optimal in variational problems. Examples are the Faber–Krahn inequality and the isoperimetric inequality. It is the $n$-dimensional, symmetric version of the decreasing rearrangement on $[0, \infty)$.