Projection body

Remarks

That this function is a support function follows from Cauchy's projection formula $\mathrm{vol}_{n-1}(K | u^\perp) = \frac{1}{2} \int_{S^{n-1}} \abs{u \cdot v} \, dS_K(v)$, where $S_K$ is the surface area measure of $K$. The formula also shows that $\Pi K$ is symmetric about the origin and is a zonoid. Example: let $C = [0, 1]^n$. For a unit vector $u$, the shadow of $C$ on $u^\perp$ has volume $\abs{u_1} + \cdots + \abs{u_n}$. This is the support function of $[-1, 1]^n$ at $u$, so $\Pi C = [-1, 1]^n$. Non-example: for $n \ge 3$, every projection of a segment $K$ has $(n-1)$-volume $0$. So the formula gives $h = 0$ and the set $\set{0}$, which is not a convex body; this is why $K$ must have nonempty interior. For the unit ball $B$, every shadow is an $(n-1)$-ball, so $\Pi B = \omega_{n-1} B$, with $\omega_{n-1}$ the volume of the unit ball of $\R^{n-1}$.