Boxing inequalities in Banach spaces
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 ${\rm HC}_m(W^{n+1})\leq c(m){\rm HC}_m(M^n)$. He...
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Main Authors | , |
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Format | Journal Article |
Language | English |
Published |
05.04.2023
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Subjects | |
Online Access | Get full text |
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Summary: | 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: 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) ${\rm
HC}_m(H(X\times [0,1]))\leq c_2(m){\rm HC}_m(X)$. This theorem strengthens the
main result of [LLNR]. |
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DOI: | 10.48550/arxiv.2304.02709 |