Intersection body

Definition

Let $n \ge 2$ , and let $K \subseteq \R^n$ be a convex body with the origin in its interior. Let $S^{n-1}$ be the unit sphere of $\R^n$ , and for $u \in S^{n-1}$ let $u^\perp = \set{x : x \cdot u = 0}$ . Write $\mathrm{vol}_{n-1}$ for the $(n-1)$ -dimensional volume in the hyperplane $u^\perp$ . The intersection body of $K$ is the set $IK = \set{r u : u \in S^{n-1}, \ 0 \le r \le \mathrm{vol}_{n-1}(K \cap u^\perp)}.$ So $IK$ is star-shaped about the origin, and its radial function is $\rho_{IK}(u) = \mathrm{vol}_{n-1}(K \cap u^\perp)$ .

Remarks

Example: let $B$ be the closed unit ball of $\R^n$ and $\omega_{n-1}$ the volume of the unit ball of $\R^{n-1}$ . Every section $B \cap u^\perp$ is a unit ball of dimension $n - 1$ , so $\rho_{IB} = \omega_{n-1}$ and $IB = \omega_{n-1} B$ . Non-example: let the origin lie outside $K$ . Then some hyperplane $u^\perp$ misses $K$ , and the radial function vanishes in the direction $u$ . So the construction no longer gives a body with the origin in its interior, which is why the definition requires the origin to be an interior point of $K$ . Busemann's theorem says that $IK$ is convex when $K$ is convex and symmetric about the origin. The definition is often stated more generally for star bodies, with the same formula for the radial function.

Reading path

To understand this page from the ground up, read these 3 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
n-sphere object
Intersection body definition

Tag	`R1S7NT139R`
URI	`m2://R1S7NT139R`
Kind	definition
Writing	W0 stub
MSC	52A20, 52A40
Defines	Intersection body
Uses	Convex body, n-sphere

Intersection body

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here