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

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 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 Lip(0)(L-n), the space of Lipschitz functions on L-n. We deduce that the Banach-Mazur distance from TC(L-n), the transportation cost space of L-n, to l(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 Lip(0)(D-n,D-k), where D-n,D-k is the diamond graph of branching k. We also provide simple examples of finite metric spaces, transportation cost spaces on which contain l(infinity)(3) and l(infinity)(4) isometrically.