Projection body

Summary

Let $K$ be a convex body in $\R^n$. Its projection body $\Pi K$ is a convex body given by its support function. At a unit vector $u$, this function is the volume of the shadow of $K$ on the hyperplane orthogonal to $u$. It packs all the shadows of $K$ into one convex body, which is always symmetric about the origin. Projection bodies were introduced by Minkowski and are central in the study of affine isoperimetric inequalities.