Analysis on Laakso graphs with application to the structure of transportation cost spaces

This article is a continuation of our article in [Canad. J. Math. Vol. 72 (3), (2020), pp. 774--804]. 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 $\operatorname{Lip}_0(\mathcal{L}_n)$, the space of Lipschitz functions on $\mathcal{L}_n$. We deduce that the Banach-Mazur distance from TC$(\mathcal{L}_n)$, 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 $\operatorname{Lip}_0(D_{n,k})$, 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.