Let ##IMG## [http://ej.iop.org/images/1064-5616/211/6/786/MSB_211_6_786ieqn4.gif] {$S \subset \mathbb{R}^{n}$} be a nonempty closed set such that for some ##IMG## [http://ej.iop.org/images/1064-5616/211/6/786/MSB_211_6_786ieqn5.gif] {${d \in [0,n]}$} and ##IMG## [http://ej.iop.org/images/1064-5616/211/6/786/MSB_211_6_786ieqn6.gif] {$\varepsilon>0$} the ##IMG## [http://ej.iop.org/images/1064-5616/211/6/786/toc_MSB_211_6_786ieqn2.gif] {$d$} -Hausdorff content ##IMG## [http://ej.iop.org/images/1064-5616/211/6/786/MSB_211_6_786ieqn7.gif] {$\mathscr{H}^{d}_{\infty}(S \cap Q(x,r)) \geq \varepsilon r^{d}$} for all cubes ##IMG## [http://ej.iop.org/images/1064-5616/211/6/786/MSB_211_6_786ieqn8.gif] {$Q(x,r)$} with centre ##IMG## [http://ej.iop.org/images/1064-5616/211/6/786/MSB_211_6_786ieqn9.gif] {$x \in S$} and edge length ##IMG##

中文翻译：

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Sobolev ## IMG ## [http://ej.iop.org/images/1064-5616/211/6/786/toc_MSB_211_6_786ieqn1.gif] {$ W ^ {1} _ {p} $}-＃上的空格＃IMG ## [http://ej.iop.org/images/1064-5616/211/6/786/toc_MSB_211_6_786ieqn2.gif] {$ d $}-## IMG ## {$的厚封闭子集。 。}

让## IMG ## [http://ej.iop.org/images/1064-5616/211/6/786/MSB_211_6_786ieqn4.gif] {$ S \ subset \ mathbb {R} ^ {n} $}为一个非空的封闭集，对于某些## IMG ## [http://ej.iop.org/images/1064-5616/211/6/786/MSB_211_6_786ieqn5.gif] {$ {d \ in [0，n ]} $}和## IMG ## [http://ej.iop.org/images/1064-5616/211/6/786/MSB_211_6_786ieqn6.gif] {$ \ varepsilon> 0 $} ## IMG＃ ＃[http://ej.iop.org/images/1064-5616/211/6/786/toc_MSB_211_6_786ieqn2.gif] {$ d $}-Hausdorff内容## IMG ## [http：//ej.iop。 org / images / 1064-5616 / 211/6/786 / MSB_211_6_786ieqn7.gif] {$ \ mathscr {H} ^ {d} _ {\ infty}（S \ cap Q（x，r））\ geq \ varepsilon r所有多维数据集的^ {d} $} ## IMG ## [http://ej.iop.org/images/1064-5616/211/6/786/MSB_211_6_786ieqn8.gif] {$ Q（x，r）$ }，中心## IMG ## [http://ej.iop.org/images/1064-5616/211/6/786/MSB_211_6_786ieqn9.gif] {$ x Sin}和边缘长度## IMG＃ ＃