# 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 $\Phi$.