BIRS Workshop Lecture Videos

Banff International Research Station Logo

BIRS Workshop Lecture Videos

Convex geometry and waist inequalities Klartag, Boaz


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 $K$.

Item Media

Item Citations and Data


Attribution-NonCommercial-NoDerivatives 4.0 International