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A sharp necessary condition for rectifiable curves in metric spaces
Revista Matemática Iberoamericana ( IF 1.2 ) Pub Date : 2020-10-06 , DOI: 10.4171/rmi/1216
Guy David 1 , Raanan Schul 2
Affiliation  

In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiable curves in the plane, using a multiscale sum of what is now known as Jones $\beta$ -numbers, numbers measuring flatness in a given scale and location. This work was generalized to $\mathbb R^n$ by Okikiolu, to Hilbert space by the second author, and has many variants in a variety of metric settings. Notably, in 2005, Hahlomaa gave a sufficient condition for a subset of a metric space to be contained in a rectifiable curve. We prove the sharpest possible converse to Hahlomaa’s theorem for doubling curves, and then deduce some corollaries for subsets of metric and Banach spaces, as well as the Heisenberg group.

中文翻译:

度量空间中可校正曲线的尖锐必要条件

P. Jones在其1990年的发明论文中,使用了现在称为Jones $ \ beta $数字的多尺度总和来表征平面中可校正曲线的子集,这些数字在给定的尺度和位置上测量平坦度。这份工作由Okikiolu推广到$ \ mathbb R ^ n $,第二作者推广到Hilbert空间,并且在各种度量标准设置中都有许多变体。值得注意的是,在2005年,Hahlomaa给出了足以将度量空间的子集包含在可校正曲线中的充分条件。我们证明了与Hahlomaa定理有关的曲线加倍的最大可能逆转,然后推导了度量空间和Banach空间的子集以及Heisenberg组的一些推论。
更新日期:2020-10-06
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