希尔伯特空间中Jordan曲线的旅行商定理
The Traveling Salesman Theorem for Jordan Curves in Hilbert Space
摘要：
给定一个度量空间$X$，$X$的分析家旅行商定理给出了包含任何子集$E\subseteq X$的最短曲线的长度与测量$E$的"平坦性“的多尺度和之间的定量关系。第一个这样的定理由Jones证明了$X = \mathbb{R}^2$，并由Okikiolu扩展到$X = \mathbb{R}^n$，而Schul证明了希尔伯特空间的类似定理，$X = H$。毕晓普后来证明，如果考虑乔丹弧，那么琼斯和奥基奥卢的结果所给出的定量关系可以变得尖锐。本文在希尔伯特空间中给出了Schul原来的旅行商定理的必要一半的充分证明，并提供了一个类似于Bishop在$\mathbb{R}^n$中的上述锐化的定理的数量关系的限制为Jordan弧时的锐化.
英文摘要：
Given a metric space $X$, an Analyst's Traveling Salesman Theorem for $X$ gives a quantitative relationship between the length of a shortest curve containing any subset $E\subseteq X$ and a multi-scale sum measuring the ``flatness'' of $E$. The first such theorem was proven by Jones for $X = \mathbb{R}^2$ and extended to $X = \mathbb{R}^n$ by Okikiolu, while an analogous theorem was proven for Hilbert space, $X = H$, by Schul. Bishop has since shown that if one considers Jordan arcs, then the quantitative relationship given by Jones' and Okikioulu's results can be sharpened. This paper gives a full proof of Schul's original necessary half of the traveling salesman theorem in Hilbert space and provides a sharpening of the theorem's quantitative relationship when restricted to Jordan arcs analogous to Bishop's aforementioned sharpening in $\mathbb{R}^n$.
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