Question

Suppose $K$ is an $n$-dimensional $C^2$ convex body in $\mathbb{R}^{n+1}$. We choose two distinct directions $z_0, z_1\in\mathbb{S^{n}}$. If $P_1$ and $P_2$ are the corresponding hyperplanes($z_0\perp P_1$ and $z_1\perp P_2$) and $K'$ is the projection of $K$ on $P_1\cap P_2$, what is the $Vol(K')$? We know the support function, and for simplicity let's suppose the body is symmetric and centered at the origin. If we just consider one hyperplane say, $P_0$, and want to compute the area of projection of $K$ on $P_0$ then the answer is $\frac{1}{2}\int_{\mathbb{S}^{n-1}}\frac{|\langle z,z_0\rangle|}{G}d\mu$ where $G$ is the Guass curvature of the boundary of $K$. I am looking for a solution of this type, possibly involving other symmetric functions of principle curvatures.

Answer 1

This is easier at least for me if we forget about the inner product.

Let $K$ be a convex body in the vector space $X$ and assume that the origin lies in the interior of $K$. If you know $K$, you know its support function $h: X^* \rightarrow \mathbb{R}$. Now let $N \subset X$ be a linear subspace and $\pi: X \rightarrow L = X/N$ the natural projection map. Then it's straightforward to check that the support function of $\pi(K) \subset L$ is simply the restriction of $h$ to $L^* = N^\perp$. Given that, it is straightforward to calculate the Gauss curvature and volume of $\pi(K)$.

Answer 2

You can find complete results of this type for $K$ an ellipsoid in

Rivin, Igor Surface area and other measures of ellipsoids. Adv. in Appl. Math. 39 (2007)

See particularly Theorem 31. You might be able to generalize to general convex bodies.