symmetrization

Symmetrization is a collection of procedures in geometry and analysis that transforms a given object—such as a set, a function, or a measure—into a more symmetric one while preserving certain quantities. The goal is to compare noisy or irregular configurations with more regular, highly symmetric ones and to derive extremal or qualitative information. Commonly preserved quantities include volume or measure, and certain integrals or norms remain unchanged or decrease in the process.

Steiner symmetrization is defined with respect to a chosen hyperplane or direction. For each line perpendicular to the chosen direction, the intersection with the set is replaced by a symmetric interval about the hyperplane having the same length as the original intersection. The resulting set has the same volume as the original one and often has a smaller surface area, an effect used to approach isoperimetric conclusions. Repeated applications in multiple directions tend to enhance symmetry and, in convex cases, drive the shape toward a ball.

Schwarz symmetrization, also known as the symmetric decreasing rearrangement, applies to nonnegative functions. The level sets {x : f(x) > t} are rearranged into balls centered at the origin with the same measure, yielding a radially symmetric and radially decreasing function f*. This rearrangement preserves Lp norms and the integral of f, and it typically reduces Dirichlet energy, underpinning many sharp inequalities such as isoperimetric and Sobolev inequalities.

Other forms, such as polarization and rearrangement with respect to group actions, expand the toolkit. Symmetrization is widely used in the calculus of variations, partial differential equations, and geometric analysis to obtain extremal configurations and comparison principles. See also rearrangement inequalities and the isoperimetric inequality.