高维Jordan曲线
Higher dimensional Jordan curves
摘要：
我们解决的问题是什么是正确的高维模拟的约旦曲线从定量的rectifiability的角度来看。更确切地说，我们表明，“拓扑稳定”集可以被用来作为覆盖对象的分析师的旅行推销员定理型定理：如果$E$是较低的$d$ -正则（在一定的适当的意义上），那么我们表明，存在一个拓扑稳定的表面$Γ$，使$E \subset Γ$和$$ \mathrm{diam}(E)^d + β^d(E) \approx \mathcal{H}^d(Γ), $$其中$β^d$是一个长期量化的曲率$E$。本文主要结果的一个推论和Hyde的一个构造，是Peter Jones TST的一个高维类似，对欧几里得空间的\textit{any}子集有效.
英文摘要：
We address the question of what is the correct higher dimensional analogue of Jordan curves from the point of view of quantitative rectifiability. More precisely, we show that 'topologically stable' sets can be used as covering objects in Analyst's Travelling Salesman Theorem-type theorems: if $E$ is lower $d$-regular (in a certain suitable sense), then we show that there exists a topologically stable surface $Γ$ so that $E \subset Γ$ and $$ \mathrm{diam}(E)^d + β^d(E) \approx \mathcal{H}^d(Γ), $$ where $β^d$ is a term quantifying the curvature of $E$. A corollary of the main result of this paper and a construction by Hyde, is a higher dimensional analogue of Peter Jones TST, valid for \textit{any} subset of Euclidean space.
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