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Gruenbaum's inequality for projections

Banff International Research Station for Mathematical Innovation and Discovery

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.

Mathematics; Convex and discrete geometry; Geometry; Discrete mathematics

video/mp4

Author affiliation: University of Alberta

2017-11-23

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/63716

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/