View source for Circular symmetrization

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" />.

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>