View source for Circular symmetrization

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A geometrical transformation of an open (closed) set  $  G $
in the plane, relative to a ray  $  \lambda $
emanating from a point  $  P $, 
onto a set  $  G  ^ {*} $
in the same plane defined as follows: 1) the intersection of  $  G  ^ {*} $
with some circle with centre at  $  P $
is either empty or is the entire circle, depending on whether the intersection of  $  G $
with the same circle is empty or the entire circle, respectively; and 2) if the intersection of  $  G $
with a circle with centre at  $  P $
has angular Lebesgue measure  $  \Phi $, 
then the intersection of  $  G  ^ {*} $
with the same circle is an open (closed) arc intersecting  $  \lambda $, 
symmetric about  $  \lambda $
and visible from  $  P $
at angle  $  \Phi $.

The above definition carries over in a natural way to the three-dimensional case (symmetrization relative to a half-plane). See also [[Symmetrization|Symmetrization]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Pólya,   G. Szegö,   "Isoperimetric inequalities in mathematical physics" , Princeton Univ. Press  (1951)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W.K. Hayman,   "Multivalent functions" , Cambridge Univ. Press  (1958)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.A. Jenkins,   "Univalent functions and conformal mapping" , Springer  (1958)</TD></TR></table>