Convex geometry and waist inequalities

TL;DR

This paper links Gromov's isoperimetry of waists with Milman's $M$-ellipsoid, proving a universal waist inequality for convex bodies and confirming Guth's conjecture in the unit cube case.

Contribution

It establishes a universal waist inequality for convex bodies, connecting geometric properties with isoperimetric principles, and confirms a conjecture by Guth for the unit cube.

Findings

Any convex body has a linear image with a waist inequality.

Confirmed Guth's conjecture for the unit cube case.

Established relations between waist inequalities and convex body characteristics.

Abstract

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 $K$.