Small separators, upper bounds for l^∞-widths, and systolic geometry

We investigate the dependence on the dimension in the inequalities that relate the volume of a closed submanifold Mⁿ⊂ RN with its l^∞-width Wl^∞ₙ₋₁(Mⁿ) defined as the infimum over all continuous maps φ:Mⁿ Kⁿ⁻¹N of supx∈ Mⁿ φ(x)-xl^∞. We prove that Wl^∞ₙ₋₁(Mⁿ)≤ const\ √n\ vol(Mⁿ)1/n, and if the codimension $N-n$ is equal to $1$, then Wl^∞ₙ₋₁(Mⁿ)≤ 3\ vol(Mⁿ)1/n. As a corollary, we prove that if Mⁿ⊂ RN is essential, then there exists a non-contractible closed curve on Mⁿ contained in a cube in RN with side length const\ √n\ vol1/n(Mⁿ) with sides parallel to the coordinate axes. If the codimension is $1$, then the side length of the cube is 12· vol1/n(Mⁿ).