This article is a continuation of our article in Dilworth et al. (Can J Math 72:774–804, 2020). We construct orthogonal bases of the cycle and cut spaces of the Laakso graph \(\mathcal {L}_n\). They are used to analyze projections from the edge space onto the cycle space and to obtain reasonably sharp estimates of the projection constant of \({\text {Lip}}_0(\mathcal {L}_n)\), the space of Lipschitz functions on \(\mathcal {L}_n\). We deduce that the Banach–Mazur distance from \({\mathrm{TC}}\quad (\mathcal {L}_n)\), the transportation cost space of \(\mathcal {L}_n\), to \(\ell _1^N\) of the same dimension is at least \((3n-5)/8\), which is the analogue of a result from [op. cit.] for the diamond graph \(D_n\). We calculate the exact projection constants of \({\text {Lip}}_0(D_{n,k})\), where \(D_{n,k}\) is the diamond graph of branching k. We also provide simple examples of finite metric spaces, transportation cost spaces on which contain \(\ell _\infty ^3\) and \(\ell _\infty ^4\) isometrically.

本文是Dilworth等人文章的延续。（Can J Math 72：774–804，2020）。我们构造循环的正交基并切开Laakso图\（\ mathcal {L} _n \）的空间。它们用于分析从边缘空间到循环空间的投影，并获得对Lipschitz空间\（{\ text {Lip}} _ 0（\ mathcal {L} _n）\）的投影常数的合理清晰估计。\（\ mathcal {L} _n \）上的函数。我们推断出Banach–Mazur距离\（{mathm {L} _n} \）\（{mathm {L} _n} \）到\（\ mathcal {L} _n \）的运输成本空间尺寸相同的\ ell _1 ^ N \）至少为\（（（3n-5）/ 8 \），类似于[op。钻石图\（D_n \）。我们计算\（{\ text {Lip}} _ 0（D_ {n，k}）\）的精确投影常数，其中\（D_ {n，k} \）是分支k的菱形图。我们还提供了简单的有限度量空间示例，其中运输成本空间等距包含\（\ ell _ \ infty ^ 3 \）和\（\ ell _ \ infty ^ 4 \）。