度量空间中的线段与凸性
Segments and Convexity in Metric Spaces
AI总结：
本文介绍d-凸性，研究度量空间中度量线段的性质，获得严格凸性的新刻画，阐述d-凸性与其他凸性的关系，并通过局部雪花形条件刻画度量线段平凡的度量空间。
AI中文摘要：
本文旨在为d-凸性提供统一介绍，并为度量空间中的度量线段与凸性贡献新的结构结果。在发展度量线段的基础理论后，我们研究其几何与拓扑性质，建立若干结构结果，证明在度量被替换为拓扑等价且有界的度量后，度量线段可实现任意闭子集。随后，我们从度量、门格（Menger）及严格凸空间的角度展开研究，基于度量线段上的序结构及其到实数空间R的等距嵌入性，获得严格凸性的新刻画。此外，我们从公理化凸性的角度阐述d-凸性与度量空间中其他凸性概念的关系。最后，我们通过“局部雪花形”条件刻画度量线段为平凡的度量空间。
英文摘要：
This paper aims both to provide a unified introduction to d-convexity and to contribute new structural results on metric segments and convexity in metric spaces. After developing the basic theory of metric segments, we study their geometric and topological properties, establishing several structural results, and show that metric segments realize arbitrary closed subsets after the metric is replaced by a topologically equivalent and bounded one. We then examine metrically, Menger, and strictly convex spaces, obtaining new characterizations of strict convexity in terms of an order structure on metric segments and their isometric embeddability into R. Additionally, we offer an exposition of the relationship between d-convexity and other notions of convexity in metric spaces from the standpoint of axiomatic convexity. Finally, we characterize metric spaces in which metric segments are trivial via a "local snowflaking" condition.
