View source for Symmetrization

The association to each object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s0917301.png" /> of an object <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s0917302.png" /> (of the same class) having some symmetry. Usually symmetrization is applied to closed sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s0917303.png" /> in a Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s0917304.png" /> (or in a space of constant curvature), and also to mappings; moreover, symmetrization is constructed so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s0917305.png" /> continuously depends on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s0917306.png" />. 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|isoperimetric inequality]].

Symmetrization relative to a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s0917307.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s0917308.png" />: For each non-empty section of a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s0917309.png" /> by a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173010.png" /> one constructs a sphere in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173011.png" /> with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173012.png" /> and the same <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173013.png" />-dimensional volume as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173014.png" />; the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173015.png" /> 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 [[#References|[2]]]). For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173016.png" /> this is Steiner symmetrization, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173017.png" /> it is Schwarz symmetrization.

Symmetrization relative to a half-space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173018.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173019.png" />: For each non-empty section of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173020.png" /> by a sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173021.png" /> with centre on the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173022.png" /> and lying in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173023.png" />, one constructs a spherical cap <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173024.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173025.png" /> is a sphere with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173026.png" />) of the same <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173027.png" />-dimensional volume as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173028.png" />; the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173029.png" /> formed by these caps is the result of the symmetrization. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173030.png" /> this is spherical symmetrization, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173031.png" /> it is circular symmetrization.

Symmetrization by displacement: For a convex set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173032.png" /> one constructs its symmetrization <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173033.png" /> relative to a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173034.png" />; the result of the symmetrization is the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173035.png" />, where addition of sets is taken as the vector sum.

Symmetrization by rolling: For a convex body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173036.png" /> its [[Support function|support function]] is averaged over parallel sections of the unit sphere; the result of symmetrization is the body <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173037.png" /> recovered from the support function thus obtained.

In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173038.png" /> 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 [[#References|[1]]], [[#References|[3]]]).

In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173039.png" /> 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 [[#References|[3]]]).

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|Symmetrization method]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W. Blaschke,   "Kreis und Kugel" , Chelsea, reprint  (1949)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Hadwiger,   "Vorlesungen über Inhalt, Oberfläche und Isoperimetrie" , Springer  (1957)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G. Pólya,   G. Szegö,   "Isoperimetric inequalities in mathematical physics" , Princeton Univ. Press  (1951)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  K. Leichtweiss,   "Konvexe Mengen" , Springer  (1979)</TD></TR></table>



====Comments====
Quite generally, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173040.png" /> is a finite group acting on a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173041.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173042.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173043.png" />, then the symmetrized version of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173044.png" /> is the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173045.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173046.png" />). The element

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173047.png" /></td> </tr></table>

is called a symmetrizer. For instance, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173048.png" /> is the symmetric group on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173049.png" /> letters and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173050.png" />, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173051.png" />-th tensor power of a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173052.png" /> (respectively, the vector space of polynomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173053.png" /> variables over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173054.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173055.png" /> acts naturally (by permuting tensor factors, respectively, by permuting the variables) and application of the idempotent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173056.png" /> to a tensor (respectively, a polynomial) gives the corresponding symmetrized tensor (respectively, symmetrized polynomial). Cf. also [[Symmetrization (of tensors)|Symmetrization (of tensors)]].

For suitable infinite groups symmetrizers are defined using integrals instead of sums.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173057.png" /> is a subgroup of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173058.png" />, one also considers alternation, i.e. application of the element

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173059.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173060.png" /> is the sign of the permutation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173061.png" />. A [[Young symmetrizer|Young symmetrizer]] is obtained by symmetrizing with respect to a [[Young subgroup|Young subgroup]] followed by alternation (with respect to a different Young subgroup, corresponding to a dual partition).

More generally, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173062.png" /> is any [[Character of a group|character of a group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173063.png" /> acting on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173064.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173065.png" /> is a subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173066.png" />, then the symmetrizer defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173067.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173068.png" /> is

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173069.png" /></td> </tr></table>

The alternator corresponds to the alternating character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173070.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091730/s09173071.png" />.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Bandle,   "Isoperimetric inequalities"  P.M. Gruber (ed.)  J.M. Wills (ed.) , ''Convexity and its applications'' , Birkhäuser  (1983)  pp. 30–48</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.G. Eggleston,   "Convexity" , Cambridge Univ. Press  (1963)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R.V. Benson,   "Euclidean geometry and convexity" , McGraw-Hill  (1966)  pp. Chapt. 6</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H. Weyl,   "The classical groups" , Princeton Univ. Press  (1946)  pp. 120</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M. Marcus,   "Finite dimensional multilinear algebra" , '''1''' , M. Dekker  (1973)  pp. 78ff</TD></TR></table>