  This paper presents 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. In the specific case where
$K = [0,1]^n$ it is shown that 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 $K$.
