---
tag: R1S7NT139R
kind: definition
name: Intersection body
msc: [52A20, 52A40]
defines: Y4ZE8FVZ8D
uses: [437FSG3419, T62369RFH6]
sources:
  - url: https://en.wikipedia.org/wiki/Busemann%E2%80%93Petty_problem
---

## Definition

Let $n \ge 2$, and let $K \subseteq \R^n$ be a [[convex body|437FSG3419]] with the origin in its interior.
Let $S^{n-1}$ be the unit [[sphere|T62369RFH6]] 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.