page_title: Symmetrization methods
section_title: Circular symmetrization
hierarchical_section_title: Symmetrization methods / Circular symmetrization
caption_attribution_description: English: Circular symmetrization of set Omega
context_page_description: In mathematics the symmetrization methods are algorithms of transforming a set to a ball with equal volume and centered at the origin. B is called the symmetrized version of A, usually denoted. 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. The conjectured answer was the disk and Steiner in 1838 showed this to be true using the Steiner symmetrization method. 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. Another problem is that the Newtonian capacity of a set A is minimized by and this was proved by Polya and G. Szego using circular symmetrization.
context_section_description: A popular method for symmetrization in the plane is Polya's circular symmetrization. After, its generalization will be described to higher dimensions. Let  be a domain; then its circular symmetrization  with regards to the positive real axis is defined as follows: Let
  i.e. contain the arcs of radius t contained in . So it is defined 
  If  is the full circle, then .
  If the length is , then .
   iff .
  In higher dimensions , its spherical symmetrization  wrt to positive axis of  is defined as follows: Let
   
  i.e. contain the caps of radius r contained in . Also, for the first coordinate let  if . So as above
  If  is the full cap, then .
  If the surface area is , then  and  where  is picked so that its surface area is . In words,  is a cap symmetric around the positive axis  with the same area as the intersection .
   iff .
caption: Circular symmetrization of set 
    
      
        
          Ω
        
      
      {\displaystyle \Omega }
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