重心胶合与稳定度量的几何
Barycentric gluing and geometry of stable metrics
摘要：
我们讨论了局部到全局的嵌入技术和稳定的度量空间及其两个重要子类：局部有限空间和适当空间的度量几何的各个方面。我们解释了如何重心胶合技术，这主要是应用于双Lipschitz嵌入问题有关的局部有限空间，可以成功地实现在更广泛的背景下。例如，我们证明了任意度量空间到$\ell_p$的可嵌入性是由其球的可嵌入性决定的。我们还引入了上稳定性的概念。这个新的度量不变量形式上介于Maurey-Krivine（度量）稳定性概念和Kalton的性质$\mathcal{Q}$之间。我们发现，雷诺和Kalton的稳定度量的几个结果可以扩展到更广泛的上下文中的上稳定度量，我们指出的相关性上稳定的长期嵌入问题提出的Kalton。压缩指数理论的应用突出，我们回顾旧的和国家新的，重要的开放问题。这篇文章是写在一个风格，有利于清晰简洁，以使材料的吸引力，访问，并可重复使用的几何学家从各种背景，而不仅仅是巴拿赫空间几何学家。
英文摘要：
We discuss various aspects of a local-to-global embedding technique and the metric geometry of stable metric spaces and of two of its important subclasses: locally finite spaces and proper spaces. We explain how the barycentric gluing technique, which has been mostly applied to bi-Lipschitz embedding problems pertaining to locally finite spaces, can be implemented successfully in a much broader context. For instance, we show that the embeddability of an arbitrary metric space into $\ell_p$ is determined by the embeddability of its balls. We also introduce the notion of upper stability. This new metric invariant lies formally between Maurey-Krivine (metric) notion of stability and Kalton's property $\mathcal{Q}$. We show that several results of Raynaud and Kalton for stable metrics can be extended to the broader context of upper stable metrics and we point out the relevance of upper stability to a long-standing embedding problem raised by Kalton. Applications to compression exponent theory are highlighted and we recall old and state new, important open problems. This article was written in a style favoring clarity over conciseness in order to make the material appealing, accessible, and reusable to geometers from a variety of backgrounds, and not only to Banach space geometers.
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