We are going to discuss the following generalization of the classical boxing inequality:Let $M^n$ be a manifold in a finite- or infinite-dimensional Banach space $B$, and $m\leq n$ a positive number.Then there exists a pseudomanifold $W^{n+1}$ in $B$ such that $\partial W^{n+1}=M^n$, and the $m$-dimensional Hausdorff content $HC_m(W^{n+1})$ of $W^{n+1}$does not exceed $c(m)HC_m(M^n)$. Recall that $HC_m(X)$ is defined as the infimum of $\Sigma_i r_i^m$ over all coverings of $X$ by metric balls in $B$, where $r_i$ denote the radii of these balls.

We will discuss further generalizations of this result, its connections with systolic geometry, and with Urysohn width-volume inequalities. (The informal meaning of the first such inequality that was proven by L. Guth is that each Riemannian manifold with a small volume must be ``close" to a polyhedron of a lower dimension. )

We will present two new $l^\infty$-width - volume inequalities and discuss their implications for systolic geometry.

Joint work with Sergey Avvakumov.