有限正定度量空间的度量度收敛性
Convergence of magnitude of finite positive definite metric spaces
AI总结：
本文研究了有限正定度量空间的度量度在Gromov-Hausdorff距离下的连续性，通过外接圆半径的几何解释提出了一种新的分析方法。
摘要：
度量空间的度量度似乎并不具备简单的、方便的连续性性质，先前的研究在附加约束或较弱的概念下给出了肯定的结果，同时也提供了反例。在此背景下，我们讨论有限正定度量空间的度量度相对于Gromov-Hausdorff距离的连续性，但其领域限制基于有限度量空间足够小邻域的自然划分。结果，本文的主要定理解释了一个度量空间的基数条件，该条件决定了度量度的连续性。本研究利用度量度的几何解释作为相应有限欧几里得子集的外接圆半径。这种转换在构造反例时尤其有用，因为我们可以依赖欧几里得几何直觉。
英文摘要：
The magnitude of metric spaces does not appear to possess a simple, convenient continuity property, and previous studies have presented affirmative results under additional constraints or weaker notions, as well as counterexamples. In this vein, we discuss the continuity of magnitude of finite positive definite metric spaces with respect to the Gromov-Hausdorff distance, but with a restriction of the domain based on a canonical partition of a sufficiently small neighborhood of a finite metric space. As a result, the main theorem of this article explains a condition on the cardinality of metric spaces that determines the continuity of magnitude. This study takes advantage of the geometric interpretation of magnitude as the circumradius of the corresponding finite Euclidean subset. Such a transformation is especially useful for constructing counterexamples, as we can depend on Euclidean geometric intuition.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
