Boxing inequalities in Banach spaces and Riemannian manifolds

Abstract

We prove the following result: For each closed $n$-dimensional manifold $M$ in a (finite or infinite-dimensional) Banach space $B$, and each positive real $m\leq n$ there exists a pseudomanifold $W^{n+1}\subset B$ such that $\partial W^{n+1}=M^n$ and ${\rm HC}_m(W^{n+1})\leq c(m){\rm HC}_m(M^n)$. Here ${\rm HC}_m(X)$ denotes the $m$-dimensional Hausdorff content, i.e the infimum of $\Sigma_i r_i^m$, where the infimum is taken over all coverings of $X$ by a finite collection of open metric balls, and $r_i$ denote the radii of these balls. In the classical case, when $B=\mathbb{R}^{n+1}$, this result implies that if $\Omega\subset R^{n+1}$ is a bounded domain, then for all $m\in (0,n]$ ${\rm HC}_m(\Omega)\leq c(m){\rm HC}_m(\partial \Omega)$. This inequality seems to be new despite being well-known and widely used in the case, when $m=n$ (Gustin's boxing inequality, [G]). The result is a corollary of the following more general theorem that strengthens a theorem in [LLNR]: For each compact subset $X$ in a Banach space $B$ and positive real number $m$ such that ${\rm HC}_m(X)\not= 0$ there exists a finite $(\lceil m\rceil-1)$-dimensional simplicial complex $K\subset B$, a continuous map $\phi:X\longrightarrow K$, and a homotopy $H:X\times [0,1]\longrightarrow B$ between the inclusion of $X$ and $\phi$ (regarded as a map into $B$) such that: (1) For each $x\in X$ $\Vert x-\phi(x)\Vert_B\leq c_1(m){\rm HC}_m^{\frac{1}{m}}(X)$; (2) ${\rm HC}_m(H(X\times [0,1]))\leq c_2(m){\rm HC}_m(X)$. A similar theorem can also be proven in the case when $B$ is a metric space with a linear contractibility function and applies to all compact sets $X$ with a controllably small ${\rm HC}_m$ in Riemannian manifolds $M^n$ with the sectional curvature bounded below, the volume bounded below by a positive number, and the diameter bounded above.