关于最小参数网络的存在性
On the Existence of Minimal Parametric Networks
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中文总结 AI 辅助
研究度量空间中最小网络存在条件，引入紧装备伪度量空间概念，证明其条件足以保证最小参数网络存在，推广相关定理，还能更一般表述贝德诺夫定理，且展示了超空间继承该性质的情况。
中文摘要 AI 辅助
本文继续研究度量空间中最小网络存在的条件。我们引入了紧装备伪度量空间的概念，其中每个闭球都配备了一个紧拓扑，相对于该拓扑，原始伪度量是下半连续的。证明了这些条件足以保证任何类型的最小参数网络的存在。这推广了伊万诺夫、特罗平和图日林关于此类网络在恰当度量空间中存在的相应定理。此外，它使我们能够更一般地表述贝德诺夫关于巴拿赫空间中最小斯坦纳树存在的充分条件的定理。我们还展示了哪些超空间以及在什么条件下继承这种紧装备性质。
英文摘要
The paper continues the investigation of conditions for the existence of minimal networks in metric spaces. We introduce the notion of a compactly equipped pseudometric space, in which every closed ball is equipped with a compact topology with respect to which the original pseudometric is lower semicontinuous. It is proved that these conditions are sufficient for the existence of minimal parametric networks of any type. This generalizes the corresponding theorems of Ivanov, Tropin and Tuzhilin on the existence of such networks in proper metric spaces. Moreover, it allows us to give a more general formulation of Bednov's theorem on sufficient conditions for the existence of minimal Steiner trees in Banach spaces. We also show which hyperspaces and under what conditions inherit this compactly equipped property.
