论文标题
Banach空间中可纠正曲线的子集I：旅行推销员定理中的尖锐指数
Subsets of rectifiable curves in Banach spaces I: sharp exponents in traveling salesman theorems
论文作者
论文摘要
分析师的旅行人员问题是在公制空间中找到可纠正曲线子集的表征。琼斯（Jones）在1990年在飞机上引入和解决了这个问题，随后由Okikiolu在1992年在较高维度的欧几里得空间中解决，并在2007年Schul的无限维持希尔伯特空间$ \ ell_2 $中，在本文中，我们在本文中建立了适用于Schul必要的条件的庞大条件。曲线从$ p = 2 $到$ 1 <p <\ infty $。虽然当$ p = 2 $时，必要和足够的条件重合，但我们证明，当$ p \ neq 2 $时，必要条件和足够条件之间存在严格的差距。我们还通过Schul确定并纠正证明中的技术错误。这项调查的一部分是由埃德伦（Edelen），纳伯（Naber）和瓦尔托塔（Valtorta）最近在Banach空间中的Reifenberg-type定理进行的，并补充了Hahlomaa的作品以及David和Schul在总体公制空间中分析师TSP的最新工作。
The Analyst's Traveling Salesman Problem is to find a characterization of subsets of rectifiable curves in a metric space. This problem was introduced and solved in the plane by Jones in 1990 and subsequently solved in higher-dimensional Euclidean spaces by Okikiolu in 1992 and in the infinite-dimensional Hilbert space $\ell_2$ by Schul in 2007. In this paper, we establish sharp extensions of Schul's necessary and sufficient conditions for a bounded set $E\subset\ell_p$ to be contained in a rectifiable curve from $p=2$ to $1<p<\infty$. While the necessary and sufficient conditions coincide when $p=2$, we demonstrate that there is a strict gap between the necessary condition and sufficient condition when $p\neq 2$. We also identify and correct technical errors in the proof by Schul. This investigation is partly motivated by recent work of Edelen, Naber, and Valtorta on Reifenberg-type theorems in Banach spaces and complements work of Hahlomaa and recent work of David and Schul on the Analyst's TSP in general metric spaces.
