Boxing inequalities in Banach spaces and Riemannian manifolds

TL;DR

This work develops a robust boxing framework for $m$-dimensional Hausdorff content in Banach spaces and geodesic manifolds, proving that closed $n$-manifolds $M^n$ embedded in a Banach space $B$ can be filled by an $(n+1)$-dimensional pseudomanifold $W^{n+1}$ with ${\rm HC}_m(W^{n+1})\le c(m){\rm HC}_m(M^n)$ for all $m\le n$, with concrete Euclidean corollaries. The core method replaces cone-based arguments with an iterative $m$-filling scheme, using dyadic cube coverings, collars, and insertions to control ${\rm HC}_m$ while constructing a homotopy from $M^n$ to a lower-dimensional complex inside $B$. A key technical contribution is a dimension-dependent boxing theorem (Theorem 1.5) and its finite-dimensional reduction, enabling a full proof of a Banach-space boxing inequality and a corresponding filling radius bound; this is extended to linearly contractible geodesic spaces and to Riemannian manifolds under curvature, volume, and injectivity-radius bounds (Theorems 1.8, 5.1–5.4). The results generalize Gustin’s boxing inequality to all $m\in(0,n]$, connect to isoperimetric inequalities in Banach spaces, and yield dimension-independent or dimension-controlled constants, with implications for Urysohn width and filling radius relations.

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 $Σ_i r_i^m$, 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 $Ω\subset R^{n+1}$ is a bounded domain, then for all $m\in (0,n]$ ${\rm HC}_m(Ω)\leq c(m){\rm HC}_m(\partial Ω)$. 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 $φ:X\longrightarrow K$, and a homotopy $H:X\times [0,1]\longrightarrow B$ between the inclusion of $X$ and $φ$ (regarded as a map into $B$) such that: (1) For each $x\in X$ $\Vert x-φ(x)\Vert_B\leq c_1(m){\rm HC}_m^{\frac{1}{m}}(X)$; (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.

Figures (11)

• Figure 1: Using insertions.
• Figure 2: We added the union of the metric balls $\beta$ covering $X_k\cap\partial B(x_i, R_i)$ to $X_k$ and denoted the result by $\tilde{X}_k$. These balls $\beta$ cover all points of $X_k$ in a $\delta$-thick metric annulus bounded by $\partial B(x_i,R_i)$ on the outside. We identify the intersection of this annulus with $\tilde{X}_k$ and $(\tilde{X}_k\cap\partial B(x_i,R_i))\times [0,1]$ by means of homeomorphisms $\alpha_i$.
• Figure 3: The part of $X_k$ inside the closed ball $B(x_i,R_i-\delta_i)$ is mapped to $x_i$. Points of $X_k$ outside of the interior of the ball $B(x_i,R_i)$ are mapped to themselves. Points of $X_k$ on the middle sphere $\partial B(x_i,R_i-\frac{\delta_i}{2})$ are mapped to $K_i$ using $\vartheta_i$. Two annuli between spheres are sliced into radial arcs, and the arcs that are both in $X_k$ and the inner annulus are mapped into the straight line segments between the images of their endpoints. The arcs that are both in $X_k$ and the outer annulus are mapped using the homotopy $H_i$.
• Figure 4: Points of $X_k$ in the outer annulus $A(x_i, R_i-\frac{\delta_i}{2}, R_i)$ are mapped using the homotopy $H=H_i$.
• Figure 5: Points of $X_k$ in the inner annulus $A(x_i, R_i-\delta_i, R_i-\frac{\delta_i}{2})$ are mapped to the cone $CK_i$.

Theorems & Definitions (35)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 1.4
• Theorem 1.5
• Conjecture 1.6
• Definition 1.7
• Theorem 1.8
• Theorem 1.9
• Theorem 1.10