运输成本空间与随机树
Transportation cost spaces and stochastic trees
AI总结：
该研究围绕有限度量空间运输成本空间的$\text{ell}_1^N$-失真核心问题展开，提出基于类树结构的部分解与新技术，得到Laakso图失真的渐近紧上界并解决相关公开问题，还证明加倍度量空间有限双曲逼近的失真一致有界。
摘要：
我们研究有限度量空间上的运输成本空间，也称为Lipschitz自由空间。我们的工作受S. Dilworth、D. Kutzarova和M. Ostrovskii提出的一个核心问题推动，即找到度量空间$M$上的一个条件，该条件等价于$M$上的运输成本空间与对应维度的$\boldsymbol{\text{ell}_1^N}$之间的Banach-Mazur距离，我们将其称为$M$的$\boldsymbol{\text{ell}_1^N}$-失真。在这方面，已有一些例子被研究，例如Naor和Schechtman（2007）研究的$N\times N$网格，以及Dilworth、Kutzarova和Ostrovskii（2020）研究的Laakso图和钻石图，后续Baudier、Gartland和Schlumprecht（2023）也对其进行了研究。我们在此提出三项主要结果。首先，我们给出了该问题与度量空间类树结构相关的部分解。为此，我们开发了一种可能带来该问题完整解的新技术，并利用它得到了Laakso图的$\text{ell}_1^N$-失真的渐近紧上界，完全解决了Dilworth、Kutzarova和Ostrovskii提出的一个公开问题。最后，我们应用我们的技术证明了加倍度量空间的有限双曲逼近具有一致有界的$\text{ell}_1^N$-失真。
英文摘要：
We study transportation cost spaces over finite metric spaces, also known as Lipschitz free spaces. Our work is motivated by a core problem posed by S. Dilworth, D. Kutzarova and M. Ostrovskii, namely, find a condition on a metric space $M$ equivalent to the Banach-Mazur distance between the transportation cost space over $M$ and $\ell_1^N$ of the corresponding dimension, which we call the $\ell_1^N$-distortion of $M$. In this regard, some examples have been studied like the $N\times N$ grid by Naor and Schechtman (2007) and the Laakso and diamond graphs by Dilworth, Kutzarova and Ostrovskii (2020), later studied by Baudier, Gartland and Schlumprecht (2023). We present here three main results. Firstly, we give a partial solution to this problem relating to the tree-like structure of the metric space. For that purpose, we develop a new technique that could potentially lead to a complete solution of the problem and utilize it to find an asymptotically tight upper bound of the $\ell_1^N$-distortion of the Laakso graphs, fully solving an open problem raised by Dilworth, Kutzarova and Ostrovskii. Finally, we apply our technique to prove that finite hyperbolic approximations of doubling metric spaces have uniformly bounded $\ell_1^N$-distortion.
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