---
tag: Y4ZE8FVZ8D
kind: concept
name: Intersection body
msc: [52A20, 52A40]
sources:
  - url: https://en.wikipedia.org/wiki/Busemann%E2%80%93Petty_problem
---

## Summary

Let $K$ be a [[convex body|437FSG3419]] with the origin in its interior.
Its intersection body is the star-shaped body whose radius in each direction $u$ is the volume of the central section of $K$ orthogonal to $u$.
It records all the hyperplane sections of $K$ through the origin in one geometric object.
Intersection bodies were introduced by Lutwak, and they are the key to the answer of the [[Busemann–Petty problem|MV58A1F52E]].

## Explanations

### L1 human

Cut a solid through its centre with a flat knife held perpendicular to a direction $u$, and measure the area of the cut.
Draw a point in direction $u$ at that distance from the centre.
Doing this for every direction draws the boundary of the intersection body.
For a round ball every central cut is the same disc, so the intersection body of a ball is again a ball.