In mathematics the symmetrization methods are algorithms of transforming a set A ⊂ R n {\displaystyle A\subset \mathbb {R} ^{n}} to a ball B ⊂ R n {\displaystyle B\subset \mathbb {R} ^{n}} with equal volume vol ( B ) = vol ( A ) {\displaystyle \operatorname {vol} (B)=\operatorname {vol} (A)} and centered at the origin. B is called the symmetrized version of A, usually denoted A ∗ {\displaystyle A^{*}} . These algorithms show up in solving the classical isoperimetric inequality problem, which asks: Given all two-dimensional shapes of a given area, which of them has the minimal perimeter (for details see Isoperimetric inequality). The conjectured answer was the disk and Steiner in 1838 showed this to be true using the Steiner symmetrization method (described below). From this many other isoperimetric problems sprung and other symmetrization algorithms. For example, Rayleigh's conjecture is that the first eigenvalue of the Dirichlet problem is minimized for the ball (see Rayleigh–Faber–Krahn inequality for details). Another problem is that the Newtonian capacity of a set A is minimized by A ∗ {\displaystyle A^{*}} and this was proved by Pólya and G. Szegő (1951) using circular symmetrization (described below).
Symmetrization If Ω ⊂ R n {\displaystyle \Omega \subset \mathbb {R} ^{n}} is measurable, then it is denoted by Ω ∗ {\displaystyle \Omega ^{*}} the symmetrized version of Ω {\displaystyle \Omega } i.e. a ball Ω ∗ := B r ( 0 ) ⊂ R n {\displaystyle \Omega ^{*}:=B_{r}(0)\subset \mathbb {R} ^{n}} such that vol ( Ω ∗ ) = vol ( Ω ) {\displaystyle \operatorname {vol} (\Omega ^{*})=\operatorname {vol} (\Omega )} . We denote by f ∗ {\displaystyle f^{*}} the symmetric decreasing rearrangement of nonnegative measurable function f and define it as f ∗ ( x ) := ∫ 0 ∞ 1 { y : f ( y ) > t } ∗ ( x ) d t {\displaystyle f^{*}(x):=\int _{0}^{\infty }1_{\{y:f(y)>t\}^{*}}(x)\,dt} , where { y : f ( y ) > t } ∗ {\displaystyle \{y:f(y)>t\}^{*}} is the symmetrized version of preimage set { y : f ( y ) > t } {\displaystyle \{y:f(y)>t\}} . The methods described below have been proved to transform Ω {\displaystyle \Omega } to Ω ∗ {\displaystyle \Omega ^{*}} i.e. given a sequence of symmetrization transformations { T k } {\displaystyle \{T_{k}\}} there is lim k → ∞ d H a ( Ω ∗ , T k ( K ) ) = 0 {\displaystyle \lim \limits _{k\to \infty }d_{Ha}(\Omega ^{*},T_{k}(K))=0} , where d H a {\displaystyle d_{Ha}} is the Hausdorff distance (for discussion and proofs see Burchard (2009))
Steiner symmetrization
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