# Symmetrization

The association to each object $F$ of an object $F ^ { * }$( of the same class) having some symmetry. Usually symmetrization is applied to closed sets $F$ in a Euclidean space $E ^ {n}$( or in a space of constant curvature), and also to mappings; moreover, symmetrization is constructed so that $F ^ { * }$ continuously depends on $F$. Symmetrization preserves some and monotonely changes other characteristics of an object. Symmetrization is used in geometry, mathematical physics and function theory for the solution of extremal problems. The first symmetrizations were introduced by J. Steiner in 1836 for a proof of an isoperimetric inequality.

Symmetrization relative to a subspace $E ^ {n - k }$ in $E ^ {n}$: For each non-empty section of a set $F$ by a subspace $E ^ {k} \perp E ^ {n - k }$ one constructs a sphere in $E ^ {k}$ with centre $E ^ {k} \cap E ^ {n - k }$ and the same $k$- dimensional volume as $F \cap E ^ {k}$; the set $F ^ { * }$ formed by these spheres is the result of the symmetrization. Symmetrization relative to a subspace preserves volume and convexity, and does not increase the area of the boundary or the integral of the transversal measure (see ). For $k = 1$ this is Steiner symmetrization, for $k = n - 1$ it is Schwarz symmetrization.

Symmetrization relative to a half-space $H ^ {n - k }$ in $E ^ {n}$: For each non-empty section of $F$ by a sphere $S ^ {k}$ with centre on the boundary $\partial H ^ {n - k }$ and lying in $E ^ {k + 1 } \perp H ^ {n - k }$, one constructs a spherical cap $S ^ {k} \cap D ^ {n}$( where $D ^ {n}$ is a sphere with centre $H ^ {n - k } \cap S ^ {k}$) of the same $k$- dimensional volume as $F \cap S ^ {k}$; the set $F ^ { * }$ formed by these caps is the result of the symmetrization. For $k = n - 1$ this is spherical symmetrization, if $n = 2$ it is circular symmetrization.

Symmetrization by displacement: For a convex set $F \subset E ^ {n}$ one constructs its symmetrization $F ^ { \prime }$ relative to a subspace $E ^ {k}$; the result of the symmetrization is the set $F ^ { * } = ( F + F ^ { \prime } )/2$, where addition of sets is taken as the vector sum.

Symmetrization by rolling: For a convex body $F \subset E ^ {n}$ its support function is averaged over parallel sections of the unit sphere; the result of symmetrization is the body $F ^ { * }$ recovered from the support function thus obtained.

In $E ^ {3}$ Steiner symmetrization preserves volume and does not increase surface area, diameter and capacity; Schwarz symmetrization preserves continuity of the Gaussian curvature of the boundary and does not reduce its minimum; symmetrization relative to a half-space does not increase the fundamental frequency of the domain or the area of the boundary; spherical symmetrization does not increase capacity; symmetrization by displacement preserves the integral of the mean curvature of the boundary and does not reduce the area of the latter; symmetrization by rolling preserves width (see , ).

In $E ^ {2}$ Steiner symmetrization does not increase the polar moment of inertia, the exterior radius, the capacity or the fundamental frequency; it does not reduce torsional rigidity or the maximal interior conformal radius (see ).

In connection with the manifold applications of symmetrization, the question of convergence of symmetrizations has been considered. The definition of the analogues of symmetrization for non-closed sets permits ramification. On the use of symmetrization in function theory see Symmetrization method.

How to Cite This Entry:
Symmetrization. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Symmetrization&oldid=48928
This article was adapted from an original article by S.L. Pecherskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article