量值的小尺度极限与单点性质
度量几何
2025-03-27 v3 一般拓扑
摘要
度量空间的量值是一个实值函数，其参数控制度量的尺度。若当空间缩小至一点时量值收敛到1，则称该度量空间具有单点性质。并非每个有限度量空间都具有单点性质：迄今为止，仅发现一个有限空间不满足该性质。理解单点性质的失效对于阐明量值的解释及其关于Gromov–Hausdorff拓扑的稳定性具有重要意义。我们证明单点性质对有限度量空间是通有的，但当它失效时，失效程度可以任意严重：量值的小尺度极限可以取大于1的任意实数值。
引用
@article{arxiv.2312.14497,
title = {The small-scale limit of magnitude and the one-point property},
author = {Emily Roff and Masahiko Yoshinaga},
journal= {arXiv preprint arXiv:2312.14497},
year = {2025}
}
备注
14 pages, 4 figures. Version 2: includes acknowledgements and an updated reference, and a few typos are corrected. Version 3: references updated; minor corrections and improvements following referee suggestions. This is the final version, to appear in the Bulletin of the London Mathematical Society