Steiner symmetrization

Definition

Let $n \ge 2$ , let $K \subseteq \R^n$ be a convex body, and let $u \in \R^n$ be a unit vector. Let $u^\perp = \set{x \in \R^n : x \cdot u = 0}$ . For $x \in u^\perp$ , the set $K \cap \set{x + t u : t \in \R}$ is empty or a closed segment; let $\ell(x)$ be its length when it is nonempty. The Steiner symmetrization of $K$ in the direction $u$ is the set $S_u K = \set{x + t u : x \in u^\perp, \ K \cap (x + \R u) \ne \emptyset, \ \abs{t} \le \tfrac{1}{2} \ell(x)}.$

Remarks

The set $S_u K$ is again a convex body, it is symmetric under the reflection in $u^\perp$ , and it has the same volume as $K$ . Its surface area and its diameter are at most those of $K$ . Example: let $K$ be the triangle with vertices $(0, 0)$ , $(2, 0)$ , $(0, 2)$ in $\R^2$ and $u = (0, 1)$ . The vertical chord over $x \in [0, 2]$ has length $2 - x$ . So $S_u K = \set{(x, t) : 0 \le x \le 2, \ \abs{t} \le (2 - x)/2}$ , the triangle with vertices $(0, 1)$ , $(0, -1)$ , $(2, 0)$ , of area $2$ like $K$ . Non-example: the set $\set{x + t u : 0 \le t \le \ell(x)}$ has the same chord lengths as $S_u K$ , but its chords are not centred on $u^\perp$ , so it is not the Steiner symmetrization. Some authors write $\mathrm{st}_u K$ for $S_u K$ ; the operation is also defined for compact sets, with $\ell(x)$ the one-dimensional measure of the section.

Reading path

To understand this page from the ground up, read these 2 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Convex body concept
Steiner symmetrization definition

Tag	`3YF7P33WRM`
URI	`m2://3YF7P33WRM`
Kind	definition
Writing	W0 stub
MSC	52A20, 52A40
Defines	Steiner symmetrization
Uses	Convex body

Steiner symmetrization

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here