Open Collections

Description

Gromov and Memarian (2003--2011) have established the \emph{waist inequality} asserting that for any continuous map $f : \mathbb S^n \to \mathbb R^{n-k}$ there exists a fiber $f^{-1}(y)$ such that every its $t$-neighborhood has measure at least the measure of the $t$-neighborhood of an equatorial subsphere $\mathbb S^{k}\subset \mathbb S^n$. Going to the limit we may say that the $(n-k)$-volume of the fiber $f^{-1}(y)$ is at least that of the standard sphere $\mathbb S^{k}$. We extend this limit statement to the exact bounds for balls in spaces of constant curvature, tori, parallelepipeds, projective spaces and other metric spaces. By the volume of preimages for a non-regular map $f$ we mean its \emph{lower Minkowski content}, some new properties of which will be also presented in the talk. Joint work with Roman Karasev and Alfredo Hubard.