volume of the projected body - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T06:38:37Z http://mathoverflow.net/feeds/question/59495 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/59495/volume-of-the-projected-body volume of the projected body MINI 2011-03-24T22:01:56Z 2011-03-26T00:59:47Z <p>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.</p> http://mathoverflow.net/questions/59495/volume-of-the-projected-body/59569#59569 Answer by Igor Rivin for volume of the projected body Igor Rivin 2011-03-25T15:07:27Z 2011-03-25T15:07:27Z <p>You can find complete results of this type for $K$ an ellipsoid in </p> <p>Rivin, Igor Surface area and other measures of ellipsoids. Adv. in Appl. Math. 39 (2007)</p> <p>See particularly Theorem 31. You might be able to generalize to general convex bodies.</p> http://mathoverflow.net/questions/59495/volume-of-the-projected-body/59624#59624 Answer by Deane Yang for volume of the projected body Deane Yang 2011-03-26T00:59:47Z 2011-03-26T00:59:47Z <p>This is easier at least for me if we forget about the inner product.</p> <p>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 $\pi(K)$.</p>