How to take shortcuts in Euclidean space: making a given set into a short quasi‐convex set
How to take shortcuts in Euclidean space: making a given set into a short quasi‐convex set
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如何在欧几里得空间中走捷径：将给定集合变成短拟凸集合
DOI：
10.1112/plms/pds005
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发表时间：
2009
影响因子：
1.8
通讯作者：
Raanan Schul
中科院分区：
文献类型：
--
作者：
Jonas Azzam;Raanan Schul
对于d维欧氏空间中给定的连通集Γ，我们构造一个连通集Γ Γ，使得两个集合具有可比的Hausdorff长度，并且集合Γ具有拟凸的性质，即Γ中的任何两个点x和y可以通过一条路径连接，所有这些路径都在Γ中，该路径的长度以x和y之间的欧氏距离的固定常数倍为界。因此，对于d维欧氏空间中的任何集合K，我们有一个如上所述的集合Γ，使得Γ具有与包含K的最短连通集相当的豪斯多夫长度。这里出现的常数仅取决于环境维度d。在Γ是Reifenberg平坦的情况下，我们的常数也与维数d无关，在这种情况下，我们的定理对无穷维希尔伯特空间中的Γ成立。这项工作与出现在计算机科学中的k-spectrum密切相关。
For a given connected set Γ in d‐dimensional Euclidean space, we construct a connected set Γ˜ ⊃ Γ such that the two sets have comparable Hausdorff length, and the set Γ˜ has the property that it is quasiconvex, that is, any two points x and y in Γ˜ can be connected via a path, all of which is in Γ˜ , which has length bounded by a fixed constant multiple of the Euclidean distance between x and y. Thus, for any set K in d‐dimensional Euclidean space, we have a set Γ˜ as above such that Γ˜ has comparable Hausdorff length to a shortest connected set containing K. Constants appearing here depend only on the ambient dimension d. In the case where Γ is Reifenberg flat, our constants are also independent of the dimension d, and in this case, our theorem holds for Γ in an infinite‐dimensional Hilbert space. This work is closely related to k‐spanners, which appears in computer science.