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.


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