Convex geometry and waist inequalities

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⊆Rn has a linear image K~⊆Rn of volume one satisfying the following waist inequality: Any continuous map f:K~→Rℓ has a fiber f−1(t) whose (n−ℓ)-dimensional volume is at least cn−ℓ, where c>0 is a universal constant. In the specific case where K=[0,1]n it is shown that one may take 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.