---
tag: 8BMM456H7G
kind: concept
name: Steiner symmetrization
msc: [52A20, 52A40]
sources:
  - url: https://en.wikipedia.org/wiki/Isoperimetric_inequality
---

## Summary

Steiner symmetrization is an operation that turns a [[convex body|437FSG3419]] into a convex body that is symmetric about a chosen hyperplane.
It slides each chord of the body orthogonal to the hyperplane along its line until the hyperplane cuts the chord in its midpoint.
The operation keeps the volume, and it does not increase the surface area or the diameter.
Repeating it in well chosen directions makes a body converge to a ball, which gives proofs of the [[isoperimetric inequality|KBX8RBZ626]] and of related extremal results.

## Explanations

### L1 human

Think of a convex body as a bundle of straight sticks, all parallel to one fixed direction.
Slide every stick along its own line until its midpoint lies on one fixed plane.
The sticks keep their lengths, so the volume does not change.
The new body is a mirror image of itself across the plane, and it is "rounder" than before.
For example, the right triangle with vertices $(0, 0)$, $(2, 0)$, $(0, 2)$, symmetrized about the $x$-axis, becomes the triangle with vertices $(0, 1)$, $(0, -1)$, $(2, 0)$, which has the same area $2$.