Small separators, upper bounds for $l^\infty$-widths, and systolic geometry
Sergey Avvakumov, Alexander Nabutovsky
TL;DR
The paper establishes dimension-dependent upper bounds for the $l^Infinity$-width $W^{l^Infinity}_{n-1}(M^n)$ of closed submanifolds $M^n\subset\mathbb{R}^N$, proving $W^{l^Infinity}_{n-1}(M^n)\le C\sqrt{n}\,vol(M^n)^{1/n}$ with an absolute constant $C$ (asymptotically $C=(2\pi/\sqrt{e})(1+o(1))$). In codimension one, a sharper bound $W^{l^Infinity}_{n-1}(M^n)\le \sqrt{3}\,vol(M^n)^{1/n}$ is obtained. The authors develop a nonlinear Federer–Fleming framework built around $\mathbb{Z}^N$-periodic foams and a $l^Infinity$-kinematic formula, complemented by small $w\mathbb{Z}^N$-periodic separators, to produce high-codimension intersections that yield low-dimensional targets for projection. These methods give corollaries for essential manifolds: a non-contractible closed curve can be found inside axis-aligned cubes of side length proportional to $vol(M^n)^{1/n}$ (with explicit constants in codimension one). The paper also connects these bounds to systolic geometry through Gromov’s systolic inequality and Guth’s UW-width bound, and presents two proofs of the main result, highlighting a robust nonlinear approach to dimensionally reducing manifolds via foam-like constructions and kinematic averaging.
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 $W^{l^\infty}_{n-1}(M^n)$ defined as the infimum over all continuous maps $φ:M^n\longrightarrow K^{n-1}\subset\mathbb{R}^N$ of $sup_{x\in M^n}\Vert φ(x)-x\Vert_{l^\infty}$. We prove that $W^{l^\infty}_{n-1}(M^n)\leq const\ \sqrt{n}\ vol(M^n)^{\frac{1}{n}}$, and if the codimension $N-n$ is equal to $1$, then $W^{l^\infty}_{n-1}(M^n)\leq \sqrt{3}\ vol(M^n)^{\frac{1}{n}}$. 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 $const\ \sqrt{n}\ vol^{\frac{1}{n}}(M^n)$ with sides parallel to the coordinate axes. If the codimension is $1$, then the side length of the cube is $4\ vol^{\frac{1}{n}}(M^n)$. 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 $l^N_\infty$ (Theorem 3.5), and introduce high-codimension analogs of optimal foams recently discovered in [KORW] and [AK].