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symmetrizations

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Symmetrizations might refer to
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 Polya and G. Szego (1951) using circular symmetrization (described below).

Symmetrizations might refer to

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 Polya and G. Szego (1951) using circular symmetrization (described below).

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