Recall the definition of the $m$th Hausdorff content of a metric space $X$: it is the infimum of $\sum d_i^m$, where
the infimum is taken over all coverings of $X$ by a finite collection of open metric balls, and $d_i$
denote the diameters of these balls.

A subset $X$ of a Banach space (of finite or infinite dimension) can always be contracted, or "filled", via a homotopy to an $(m-1)$-dimensional subpolyhedron. The boxing inequality states that the $m$th Hausdorff content of the homotopy can be bounded in terms of the $m$th Hausdorff content of $X$. Note, that this differs from the classical isoperimetric inequality where the $(m+1)$th volume of the filling is bounded.

I will discuss the proof of this result and its connections to the notions of Gromov's filling radius, Urysohn width, systole, relations between these notions, and corresponding geometric inequalities.