Small separators, upper bounds for $l^{\infty }$-widths, and systolic geometry

Abstract

We investigate the dependence on the dimension in the inequalities that relate the Euclidean volume of a closed submanifold $M^n\subset {\mathbb{R}}^N$ with its $l^{\infty }$-width $W_{n-1}^{l^{\infty }}(M^n)$ defined as the infimum over all continuous maps $\ddot{I}•:M^n⟶K^{n-1}\subset {\mathbb{R}}^N$ of $su{p}_{x\in M^n}\Vert \ddot{I}•(x)-x{\Vert }_{l^{\infty }}$. We prove that $W_{n-1}^{l^{\infty }}(M^n)\le const\sqrt{n}vol(M^n{)}^{\frac{1}{n}}$, and if the codimension $N-n$ is equal to $1$, then $W_{n-1}^{l^{\infty }}(M^n)\le \sqrt{3}vol(M^n{)}^{\frac{1}{n}}$. As a corollary, we prove that if $M^n\subset {\mathbb{R}}^N$ is {\it essential}, then there exists a non-contractible closed curve on $M^n$ contained in a cube in ${\mathbb{R}}^N$ with side length $const\sqrt{n}vo{l}^{\frac{1}{n}}(M^n)$ with sides parallel to the coordinate axes. If the codimension is $1$, then the side length of the cube is $4vo{l}^{\frac{1}{n}}(M^n)$. To prove these results we introduce a new approach to systolic geometry that can be described as a non-linear version of the classical Federer-Fleming argument, where we push out from a specially constructed non-linear $(N-n)$-dimensional complex in ${\mathbb{R}}^N$ that does not intersect $M^n$. To construct these complexes we first prove a version of kinematic formula where one averages over isometries of $l_{\infty }^N$ (Theorem 3.5), and introduce high-codimension analogs of optimal foams recently discovered in [KORW] and [AK].