Open Collections

Description

Let $K$ be a convex body in $\mathbb{R}^n$ whose centroid is at the origin, let $E\in G(n,k)$ be a subspace, and let $\xi\in S^{n-1}$. I will discuss my joint work with Ning Zhang, where we found the best constant $c = \left(\frac{k}{n+1}\right)^k$ so that $|(K|E)\cap\xi^+|_k\geq c\,|K|E|_k$. Here, $|\cdot|_k$ is $k$-dimensional volume, $K|E$ is the projection of $K$ onto $E$, and $\xi^+ = \{ x\in\mathbb{R}^n:\, \langle x,\xi\rangle\geq 0\}$. Our result generalizes both Gr\"unbaum's inequality, and an old inequality of Minkowski and Radon.