Symmetrization methods

In mathematics the symmetrization methods are algorithms of transforming a set ${\displaystyle A\subset \mathbb {R} ^{n}}$ to a ball ${\displaystyle B\subset \mathbb {R} ^{n}}$ with equal volume ${\displaystyle \operatorname {vol} (B)=\operatorname {vol} (A)}$ and centered at the origin. B is called the symmetrized version of A, usually denoted ${\displaystyle A^{*}}$. 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 ${\displaystyle A^{*}}$ and this was proved by Polya and G. Szego (1951) using circular symmetrization (described below).

Symmetrization

If ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$ is measurable, then it is denoted by ${\displaystyle \Omega ^{*}}$ the symmetrized version of ${\displaystyle \Omega }$ i.e. a ball ${\displaystyle \Omega ^{*}:=B_{r}(0)\subset \mathbb {R} ^{n}}$ such that ${\displaystyle \operatorname {vol} (\Omega ^{*})=\operatorname {vol} (\Omega )}$. We denote by ${\displaystyle f^{*}}$ the symmetric decreasing rearrangement of nonnegative measurable function f and define it as ${\displaystyle f^{*}(x):=\int _{0}^{\infty }1_{\{y:f(y)>t\}^{*}}(x)\,dt}$, where ${\displaystyle \{y:f(y)>t\}^{*}}$ is the symmetrized version of preimage set ${\displaystyle \{y:f(y)>t\}}$. The methods described below have been proved to transform ${\displaystyle \Omega }$ to ${\displaystyle \Omega ^{*}}$ i.e. given a sequence of symmetrization transformations ${\displaystyle \{T_{k}\}}$ there is ${\displaystyle \lim \limits _{k\to \infty }d_{Ha}(\Omega ^{*},T_{k}(K))=0}$, where ${\displaystyle d_{Ha}}$ is the Hausdorff distance (for discussion and proofs see Burchard (2009) harvtxt error: no target: CITEREFBurchard2009 (help))

Steiner symmetrization

Steiner Symmetrization of set ${\displaystyle \Omega }$

Steiner symmetrization was introduced by Steiner (1838) to solve the isoperimetric theorem stated above. Let ${\displaystyle H\subset \mathbb {R} ^{n}}$ be a hyperplane through the origin. Rotate space so that ${\displaystyle H}$ is the ${\displaystyle x_{n}=0}$ (${\displaystyle x_{n}}$ is the nth coordinate in ${\displaystyle \mathbb {R} ^{n}}$) hyperplane. For each ${\displaystyle x\in H}$ let the perpendicular line through ${\displaystyle x\in H}$ be ${\displaystyle L_{x}=\{x+ye_{n}:y\in \mathbb {R} \}}$. Then by replacing each ${\displaystyle \Omega \cap L_{x}}$ by a line centered at H and with length ${\displaystyle |\Omega \cap L_{x}|}$ we obtain the Steiner symmetrized version.

${\displaystyle \operatorname {St} (\Omega ):=\{x+ye_{n}:x+ze_{n}\in \Omega {\text{ for some }}z{\text{ and }}|y|\leq {\frac {1}{2}}|\Omega \cap L_{x}|\}.}$

It is denoted by ${\displaystyle \operatorname {St} (f)}$ the Steiner symmetrization wrt to ${\displaystyle x_{n}=0}$ hyperplane of nonnegative measurable function ${\displaystyle f:\mathbb {R} ^{d}\to \mathbb {R} }$ and for fixed ${\displaystyle x_{1},\ldots ,x_{n-1}}$ define it as

${\displaystyle St:f(x_{1},\ldots ,x_{n-1},\cdot )\mapsto (f(x_{1},\ldots ,x_{n-1},\cdot ))^{*}.}$

Properties

• It preserves convexity: if ${\displaystyle \Omega }$ is convex, then ${\displaystyle St(\Omega )}$ is also convex.
• It is linear: ${\displaystyle St(x+\lambda \Omega )=St(x)+\lambda St(\Omega )}$.
• Super-additive: ${\displaystyle St(K)+St(U)\subset St(K+U)}$.

Circular symmetrization

Circular symmetrization of set ${\displaystyle \Omega }$

A popular method for symmetrization in the plane is Polya's circular symmetrization. After, its generalization will be described to higher dimensions. Let ${\displaystyle \Omega \subset \mathbb {C} }$ be a domain; then its circular symmetrization ${\displaystyle \operatorname {Circ} (\Omega )}$ with regards to the positive real axis is defined as follows: Let

${\displaystyle \Omega _{t}:=\{\theta \in [0,2\pi ]:te^{i\theta }\in \Omega \}}$

i.e. contain the arcs of radius t contained in ${\displaystyle \Omega }$. So it is defined

• If ${\displaystyle \Omega _{t}}$ is the full circle, then ${\displaystyle \operatorname {Circ} (\Omega )\cap \{|z|=t\}:=\{|z|=t\}}$.
• If the length is ${\displaystyle m(\Omega _{t})=\alpha }$, then ${\displaystyle \operatorname {Circ} (\Omega )\cap \{|z|=t\}:=\{te^{i\theta }:|\theta |<{\frac {\alpha }{2}}\}}$.
• ${\displaystyle 0,\infty \in \operatorname {Circ} (\Omega )}$ iff ${\displaystyle 0,\infty \in \Omega }$.

In higher dimensions ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$, its spherical symmetrization ${\displaystyle Sp^{n}(\Omega )}$ wrt to positive axis of ${\displaystyle x_{1}}$ is defined as follows: Let ${\displaystyle \Omega _{r}:=\{x\in \mathbb {S} ^{n-1}:rx\in \Omega \}}$ i.e. contain the caps of radius r contained in ${\displaystyle \Omega }$. Also, for the first coordinate let ${\displaystyle \operatorname {angle} (x_{1}):=\theta }$ if ${\displaystyle x_{1}=rcos\theta }$. So as above

• If ${\displaystyle \Omega _{r}}$ is the full cap, then ${\displaystyle Sp^{n}(\Omega )\cap \{|z|=r\}:=\{|z|=r\}}$.
• If the surface area is ${\displaystyle m_{s}(\Omega _{t})=\alpha }$, then ${\displaystyle Sp^{n}(\Omega )\cap \{|z|=r\}:=\{x:|x|=r}$ and ${\displaystyle 0\leq \operatorname {angle} (x_{1})\leq \theta _{\alpha }\}=:C(\theta _{\alpha })}$ where ${\displaystyle \theta _{\alpha }}$ is picked so that its surface area is ${\displaystyle m_{s}(C(\theta _{\alpha })=\alpha }$. In words, ${\displaystyle C(\theta _{\alpha })}$ is a cap symmetric around the positive axis ${\displaystyle x_{1}}$ with the same area as the intersection ${\displaystyle \Omega \cap \{|z|=r\}}$.
• ${\displaystyle 0,\infty \in Sp^{n}(\Omega )}$ iff ${\displaystyle 0,\infty \in \Omega }$.

Polarization

Polarization of set ${\displaystyle \Omega }$

Let ${\displaystyle \Omega \subset \mathbb {R} ^{n}}$ be a domain and ${\displaystyle H^{n-1}\subset \mathbb {R} ^{n}}$ be a hyperplane through the origin. Denote the reflection across that plane to the positive halfspace ${\displaystyle \mathbb {H} ^{+}}$ as ${\displaystyle \sigma _{H}}$ or just ${\displaystyle \sigma }$ when it is clear from the context. Also, the reflected ${\displaystyle \Omega }$ across hyperplane H is defined as ${\displaystyle \sigma \Omega }$. Then, the polarized ${\displaystyle \Omega }$ is denoted as ${\displaystyle \Omega ^{\sigma }}$ and defined as follows

• If ${\displaystyle x\in \Omega \cap \mathbb {H} ^{+}}$, then ${\displaystyle x\in \Omega ^{\sigma }}$.
• If ${\displaystyle x\in \Omega \cap \sigma (\Omega )\cap \mathbb {H} ^{-}}$, then ${\displaystyle x\in \Omega ^{\sigma }}$.
• If ${\displaystyle x\in (\Omega \setminus \sigma (\Omega ))\cap \mathbb {H} ^{-}}$, then ${\displaystyle \sigma x\in \Omega ^{\sigma }}$.

In words, ${\displaystyle (\Omega \setminus \sigma (\Omega ))\cap \mathbb {H} ^{-}}$ is simply reflected to the halfspace ${\displaystyle \mathbb {H} ^{+}}$. It turns out that this transformation can approximate the above ones (in the Hausdorff distance) (see Brock & Solynin (2000)).

References

• Morgan, Frank (2009). "Symmetrization". Retrieved November 2015. Check date values in: |accessdate= (help)
• Burchard, Almut (2009). "A Short Course on Rearrangement Inequalities" (PDF). Retrieved November 2015. Check date values in: |accessdate= (help)

• Kojar, Tomas (2015). "Brownian Motion and Symmetrization". arXiv:1505.01868.

• Brock, Friedemann; Solynin, Alexander (2000), "An approach to symmetrization via polarization.", Transactions of the American Mathematical Society, 352: 1759–1796, doi:10.1090/S0002-9947-99-02558-1, MR 1695019