Subsets of rectifiable curves in Hilbert space-the analyst’s TSP
Subsets of rectifiable curves in Hilbert space-the analyst’s TSP
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希尔伯特空间中的可整流曲线子集 - 分析师的 TSP
DOI：
10.1007/s11854-008-0011-y
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
Raanan Schul
中科院分区：
文献类型：
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作者：
Raanan Schul
我们研究Hilbert空间中的一维集合（Hausdorff维）。目的是对包含在有限豪斯多夫长度的连通集中的希尔伯特空间子集进行分类。我们通过推广和改进Peter Jones和Kate Okikiolu关于集合的结果来做到这一点。他们的结果构成了定量可纠偏性的基础。我们证明了以下命题的一个定量版本：有限Hausdorff长度的连通集（或一个连通集的子集）的特征是：在集合的大多数点周围的大多数尺度的内球，该集合靠近直线段（这取决于球）。这是通过一个量来完成的，类似于[Jon90]中引入的量，它是正方形函数的几何模拟。这使我们可以得出结论，对于给定的集合K，这个量的l2范数（它是K的函数）具有与包含K的最短（Hausdorff长度）连通集相当的大小。特别是，我们的结果表明，通过对定理的正确重新表述，[Jon90， Oki92]中的估计与环境维无关。
We study one dimensional sets (Hausdorff dimension) lying in a Hilbert space. The aim is to classify subsets of Hilbert spaces that are contained in a connected set of finite Hausdorff length. We do so by extending and improving results of Peter Jones and Kate Okikiolu for sets in ℝd. Their results formed the basis of quantitative rectifiability in ℝd. We prove a quantitative version of the following statement: a connected set of finite Hausdorff length (or a subset of one), is characterized by the fact that inside balls at most scales aroundmost points of the set, the set lies close to a straight line segment (which depends on the ball). This is done via a quantity, similar to the one introduced in [Jon90], which is a geometric analogue of the Square function. This allows us to conclude that for a given set K, the ℓ2 norm of this quantity (which is a function of K) has size comparable to a shortest (Hausdorff length) connected set containing K. In particular, our results imply that, with a correct reformulation of the theorems, the estimates in [Jon90, Oki92] are independent of the ambient dimension.