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Convex geometry and waist inequalities

Banff International Research Station for Mathematical Innovation and Discovery

We will discuss connections between Gromov's work on isoperimetry of waists and Milman's work on the $M$-ellipsoid of a convex body. It is proven that any convex body $K \subseteq \mathbb{R}^n$ has a linear image $\tilde{K} \subseteq \mathbb{R} ^n$ of volume one satisfying the following waist inequality: Any continuous map $f:\tilde{K} \rightarrow \mathbb{R}^{\ell}$ has a fiber $f^{-1}(t)$ whose $(n-\ell)$-dimensional volume is at least $c^{n-\ell}$, where $c > 0$ is a universal constant. Already in the case where f is linear, this constitutes a slight improvement over known results. In the specific case where $K = [0,1]^n$, one may take $\tilde{K} = K$ and $c = 1$, confirming a conjecture by Guth. We furthermore exhibit relations between waist inequalities and various geometric characteristics of the convex body $K$.

Mathematics; Convex and discrete geometry; Geometry; Discrete mathematics

video/mp4

Author affiliation: Tel Aviv University

2017-11-20

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/63648

Unreviewed

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