Symmetrization methods

If \Omega\subset \mathbb{R}^n is measurable, then it is denoted by \Omega^{*} the symmetrized version of \Omega i.e. a ball \Omega^{*}:=B_r(0)\subset\mathbb{R}^n such that \operatorname{vol}(\Omega^{*})=\operatorname{vol}(\Omega). We denote by f^{*} the symmetric decreasing rearrangement of nonnegative measurable function f and define it as f^{*}(x):=\int_0^\infty 1_{\{y:f(y)>t\}^{*}}(x) \, dt, where \{y:f(y)>t\}^{*} is the symmetrized version of preimage set \{y:f(y)>t\}. The methods described below have been proved to transform \Omega to \Omega^{*} i.e. given a sequence of symmetrization transformations \{T_k\} there is \lim\limits_{k\to \infty}d_{Ha}(\Omega^{*}, T_k(K) )=0, where d_{Ha} is the Hausdorff distance (for discussion and proofs see ) ==Steiner symmetrization==

Steiner symmetrization was introduced by Steiner (1838) to solve the isoperimetric theorem stated above. Let H\subset\mathbb{R}^n be a hyperplane through the origin. Rotate space so that H is the x_n=0 (x_n is the nth coordinate in \mathbb{R}^n) hyperplane. For each x\in H let the perpendicular line through x\in H be L_x = \{x+ye_n:y\in \mathbb{R}\}. Then by replacing each \Omega\cap L_x by a line centered at H and with length |\Omega\cap L_x| we obtain the Steiner symmetrized version. : \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 \operatorname{St}(f) the Steiner symmetrization wrt to x_n=0 hyperplane of nonnegative measurable function f:\mathbb{R}^d\to \mathbb{R} and for fixed x_1,\ldots,x_{n-1} define it as : St: f(x_1,\ldots,x_{n-1},\cdot)\mapsto (f(x_1,\ldots,x_{n-1},\cdot))^{*}. Properties • It preserves convexity: if \Omega is convex, then St(\Omega) is also convex. • It is linear: St(x+\lambda \Omega)=St(x)+\lambda St(\Omega). • Super-additive: St(K)+St(U)\subset St(K+U). ==Circular symmetrization==

A popular method for symmetrization in the plane is Polya's circular symmetrization. After, its generalization will be described to higher dimensions. Let \Omega\subset \mathbb{C} be a domain; then its circular symmetrization \operatorname{Circ}(\Omega) with regard to the positive real axis is defined as follows: Let \Omega_t:=\{\theta \in [0,2\pi]:te^{i\theta}\in \Omega\} i.e. contain the arcs of radius t contained in \Omega. So it is defined • If \Omega_t is the full circle, then \operatorname{Circ}(\Omega)\cap \{|z|=t\}:=\{|z|=t\} . • If the length is m(\Omega_t)=\alpha, then \operatorname{Circ}(\Omega)\cap \{|z|=t\}:=\{te^{i\theta}: |\theta|. • 0,\infty\in \operatorname{Circ}(\Omega) iff 0,\infty \in \Omega. In higher dimensions \Omega\subset \mathbb{R}^n, its spherical symmetrization Sp^n(\Omega) wrt to positive axis of x_1 is defined as follows: Let \Omega_r:=\{x\in \mathbb{S}^{n-1}: rx\in \Omega\} i.e. contain the caps of radius r contained in \Omega. Also, for the first coordinate let \operatorname{angle}(x_1):=\theta if x_1=rcos\theta. So as above • If \Omega_r is the full cap, then Sp^n(\Omega)\cap \{|z|=r\}:=\{|z|=r\}. • If the surface area is m_s(\Omega_t)=\alpha, then Sp^n(\Omega)\cap \{|z|=r\}:=\{x:|x|=r and 0\leq \operatorname{angle}(x_1)\leq \theta_\alpha\}=:C(\theta_\alpha) where \theta_\alpha is picked so that its surface area is m_s (C(\theta_\alpha)=\alpha. In words, C(\theta_\alpha) is a cap symmetric around the positive axis x_1 with the same area as the intersection \Omega\cap \{|z|=r\}. • 0,\infty\in Sp^n(\Omega) iff 0,\infty \in \Omega. ==Polarization==

Let \Omega\subset\mathbb{R}^n be a domain and H^{n-1}\subset\mathbb{R}^n be a hyperplane through the origin. Denote the reflection across that plane to the positive halfspace \mathbb{H}^{+} as \sigma_H or just \sigma when it is clear from the context. Also, the reflected \Omega across hyperplane H is defined as \sigma \Omega. Then, the polarized \Omega is denoted as \Omega^\sigma and defined as follows • If x\in \Omega\cap \mathbb{H}^{+}, then x\in \Omega^{\sigma}. • If x\in \Omega\cap \sigma(\Omega) \cap \mathbb{H}^{-}, then x\in \Omega^{\sigma}. • If x\in (\Omega\setminus \sigma(\Omega)) \cap \mathbb{H}^{-}, then \sigma x\in \Omega^{\sigma}. In words, (\Omega\setminus \sigma(\Omega)) \cap \mathbb{H}^{-} is simply reflected to the halfspace \mathbb{H}^{+}. It turns out that this transformation can approximate the above ones (in the Hausdorff distance) (see ). ==References==