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A geometrical transformation of an open (closed) set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022320/c0223201.png" /> in the plane, relative to a ray <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022320/c0223202.png" /> emanating from a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022320/c0223203.png" />, onto a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022320/c0223204.png" /> in the same plane defined as follows: 1) the intersection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022320/c0223205.png" /> with some circle with centre at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022320/c0223206.png" /> is either empty or is the entire circle, depending on whether the intersection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022320/c0223207.png" /> with the same circle is empty or the entire circle, respectively; and 2) if the intersection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022320/c0223208.png" /> with a circle with centre at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022320/c0223209.png" /> has angular Lebesgue measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022320/c02232010.png" />, then the intersection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022320/c02232011.png" /> with the same circle is an open (closed) arc intersecting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022320/c02232012.png" />, symmetric about <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022320/c02232013.png" /> and visible from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022320/c02232014.png" /> at angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022320/c02232015.png" />.
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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]].
 
The above definition carries over in a natural way to the three-dimensional case (symmetrization relative to a half-plane). See also [[Symmetrization|Symmetrization]].

Latest revision as of 16:44, 4 June 2020


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=13034
This article was adapted from an original article by I.P. Mityuk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article