Defining Schwars rearrangement for other kind of symmetric domains

Question

Given a Lebesgue measurable subset $E \subset \mathbb{R}^N$ we denote its $N-$dimensional Lebesgue measure by $|E|$. Let $\Omega \subset \mathbb{R}^N$ a bounded measurable set and $u : \Omega \to \mathbb{R}$ be a measurable function. For $t \in \mathbb{R}$ we define $$ \{ u > t\} = \{x \in \Omega : u(x) > t\}. $$ The distribution function of $u$ is given by $$ \mu_u(t) = |\{u > t\}|. $$ The (uni dimensional) decreasing rearrangement of $u$ is defined on $[0, |\Omega|]$ by \begin{equation} u^{\#}(s) = \left\{ \begin{array}{rcll} \text{ess sup} (u),& \text{ if }s = 0 \\ \inf\{t : \mu_u(t) < s\} ,& \text{ if } 0 < s < |\Omega| \end{array}, \right. \end{equation} Given $E \subset \mathbb{R}^N$ a measurable set of finite measure. We define $E^\ast$, the symmetric rearrangement of $E$ to be $$ B_R(0), R = \left(\frac{|E|}{|B|}\right)^{\frac{1}{N}}. $$ Notice that $|E| = |E^\ast|$. Let $\Omega \subset \mathbb{R}^N$ be a bounded domain and Let $u : \Omega \to \mathbb{R}$ be a measurable function. Its Schwarz symmetrization or, the spherically symmetric and decreasing rearrangement is the function $u^\ast : \Omega^\ast \to \mathbb{R}$ defined by $$ u^\ast (x) = u^{\#}(|B||x|^N), \quad x \in \Omega^{\ast}. $$ I would like to know if is it possible to define Schwarz rearrangment with other kind of symmetriztion intead of a ball, for example, taking $E^{\ast}$ is a annulos with the same measure as $E$, in such a way that all the properties are preserved.