欧几里得空间中几类单开区域上的度量拓扑
Metric topologies over some categories of simple open regions in Euclidean space
摘要：
形状不断变化意味着什么？在凸区域的空间上，只有一个“合理”的答案。然而，在更广泛的区域类别上，例如星形区域类别，可以有许多不同的连续形状变化的“合理”定义。本文研究了n维欧几里得空间中若干有限类开有界区域上由若干度量诱导的拓扑之间的关系。具体来说，我们考虑一个同胚为基础的度量;豪斯多夫度量;双豪斯多夫度量;对称差度量;和家庭的Wasserstein度量;和拓扑，它们诱导的凸区域的空间;凸区域的空间和工会的两个分离的凸区域;和空间的星形区域。我们证明：在凸区域的空间上，所有五个度量，以及实际上满足两个一般良凸性约束的任何度量，都导出相同的拓扑。在凸区域和两个分离凸区域的并集的空间上，这五个度量都按“严格比”关系排序。按精细度降序排列，它们是：基于同胚的、对偶豪斯多夫的、豪斯多夫的、沃瑟斯坦的和对称差。此外，Wasserstein度量在它们之间是严格有序的。在星形区域的空间上，由Hausdorff度量、差分度量和Wasserstein度量导出的拓扑在精细度方面是不可比拟的。
英文摘要：
What does it mean for a shape to change continuously? Over the space of convex regions, there is only one "reasonable" answer. However, over a broader class of regions, such as the class of star-shaped regions, there can be many different "reasonable" definitions of continuous shape change. We consider the relation between topologies induced by a number of metrics over a number of limited categories of open bounded regions in n-dimensional Euclidean space. Specifically, we consider a homeomorphism-based metric; the Hausdorff metric; the dual-Hausdorff metric; the symmetric difference metric; and the family of Wasserstein metrics; and the topologies that they induce over the space of convex regions; the space of convex regions and unions of two separated convex regions; and the space of star-shaped regions. We demonstrate that: Over the space of convex regions, all five metrics, and indeed any metric that satisfies two general well-behavedness constraints, induce the same topology. Over the space of convex regions and unions of two separated convex regions, these five metrics are all ordered by "strictly finer than" relations. In descending order of fineness, these are: the homeomorphism-based, the dual-Hausdorff, the Hausdorff, the Wasserstein, and the symmetric difference. Also, Wasserstein metrics are strictly ordered among themselves. Over the space of star-shaped regions, the topologies induced by the Hausdorff metric, the symmetric-difference metric, and the Wasserstein metrics are incomparable in terms of fineness.
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