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

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 $ϕ:M^n\longrightarrow K^{n-1}\subset\mathbb{R}^N$