Intersection body

Summary

Let $K$ be a convex body 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.