Lower and upper bounds for the waists of different spaces

TL;DR

This paper advances the understanding of Gromov's waist theorem by providing new proofs and extending waist-type results to various geometric spaces using Hausdorff measure.

Contribution

It offers a simplified proof of Vaaler's theorem and extends waist bounds to projective spaces, tori, and convex bodies with Hausdorff measure considerations.

Findings

Simplified proof of Vaaler's theorem using Borsuk--Ulam--Crofton technique

Waist bounds established for real and complex projective spaces

Results extended to flat tori and convex bodies in Euclidean space

Abstract

We prove several new results around Gromov's waist theorem. We give a simple proof of Vaaler's theorem on sections of the unit cube using the Borsuk--Ulam--Crofton technique. We consider waists of real and complex projective spaces, flat tori, convex bodies in Euclidean space. We establish waist-type results in terms of the Hausdorff measure.