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 [2]). 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 [1], [3]).

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 [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.

References

[1]	W. Blaschke, "Kreis und Kugel" , Chelsea, reprint (1949) MR0076364 Zbl 0041.08802
[2]	H. Hadwiger, "Vorlesungen über Inhalt, Oberfläche und Isoperimetrie" , Springer (1957) MR0102775 Zbl 0078.35703
[3]	G. Pólya, G. Szegö, "Isoperimetric inequalities in mathematical physics" , Princeton Univ. Press (1951) MR0043486 Zbl 0044.38301
[4]	K. Leichtweiss, "Konvexe Mengen" , Springer (1979) MR0586235 MR0559138 Zbl 0442.52001 Zbl 0427.52001

Comments

Quite generally, if $ G $ is a finite group acting on a vector space $ V $ over a field $ k $, and $ v \in V $, then the symmetrized version of $ v $ is the element $ \sum _ {s \in G } sv $( or $ ( 1/ | G | ) \sum _ {s} sv $). The element

$$ e = \frac{1}{| G | } \sum _ { s } s \in k[ G] $$

is called a symmetrizer. For instance, if $ G = S _ {m} $ is the symmetric group on $ m $ letters and $ V = \otimes ^ {m} W $, the $ m $- th tensor power of a vector space $ W $( respectively, the vector space of polynomials in $ m $ variables over $ k $), then $ G $ acts naturally (by permuting tensor factors, respectively, by permuting the variables) and application of the idempotent $ e $ to a tensor (respectively, a polynomial) gives the corresponding symmetrized tensor (respectively, symmetrized polynomial). Cf. also Symmetrization (of tensors).

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

If $ G $ is a subgroup of an $ S _ {m} $, one also considers alternation, i.e. application of the element

$$ f = \frac{1}{| G | } \sum _ {\sigma \in G } \mathop{\rm sgn} ( \sigma ) \sigma , $$

where $ \mathop{\rm sgn} ( \sigma ) $ is the sign of the permutation $ \sigma $. A Young symmetrizer is obtained by symmetrizing with respect to a Young subgroup followed by alternation (with respect to a different Young subgroup, corresponding to a dual partition).

More generally, if $ \chi $ is any character of a group $ G $ acting on $ V $, and $ H $ is a subgroup of $ G $, then the symmetrizer defined by $ \chi $ and $ H $ is

$$ \frac{1}{| H | } \sum _ {h \in H } \chi ( h) h. $$

The alternator corresponds to the alternating character $ \sigma \rightarrow \mathop{\rm sgn} ( \sigma ) $ of $ S _ {m} $.

References

[a1]	C. Bandle, "Isoperimetric inequalities" P.M. Gruber (ed.) J.M. Wills (ed.) , Convexity and its applications , Birkhäuser (1983) pp. 30–48 MR0731105 Zbl 0519.53037
[a2]	H.G. Eggleston, "Convexity" , Cambridge Univ. Press (1963) MR0410561 MR1530235 MR1530200 MR0124813 MR0125931 Zbl 0331.52003 Zbl 0086.15302 Zbl 0083.38102
[a3]	R.V. Benson, "Euclidean geometry and convexity" , McGraw-Hill (1966) pp. Chapt. 6 MR0209949 Zbl 0187.44103
[a4]	H. Weyl, "The classical groups" , Princeton Univ. Press (1946) pp. 120 MR1488158 MR0000255 Zbl 1024.20502
[a5]	M. Marcus, "Finite dimensional multilinear algebra" , 1 , M. Dekker (1973) pp. 78ff MR0352112 Zbl 0284.15024