Abstract: 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 in an n-dimensional Euclidean space has a linear image K_1 of volume one satisfying the following waist inequality: Any continuous map f from K_1 to R^d has a fiber f^{-1}(t) whose (n-d)-dimensional volume is at least c^{n-d}, 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 K_1=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.

## Date:

Wed, 02/05/2018 - 12:00 to 13:00

## Location:

room 70, Ross Building