# volume of the projected body

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.

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What sort of answer are you looking for? –  JBL Mar 24 '11 at 22:56
Elaborating on JBL: This can't be determined from just what is given (which is essentially no information about $K$). Could you state what information is known about the body in question? For instance, do we know it's support function? –  Logan Maingi Mar 24 '11 at 23:56

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 $\pi(K)$.

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