# Circular symmetrization

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.

#### References

[1] | G. Pólya, G. Szegö, "Isoperimetric inequalities in mathematical physics" , Princeton Univ. Press (1951) |

[2] | W.K. Hayman, "Multivalent functions" , Cambridge Univ. Press (1958) |

[3] | J.A. Jenkins, "Univalent functions and conformal mapping" , Springer (1958) |

**How to Cite This Entry:**

Circular symmetrization.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Circular_symmetrization&oldid=46346