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

Abstract

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 ${\mathrm{Lip}}_0({\mathcal{L}}_n)$, 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 ${\mathrm{Lip}}_0(D_{n,k})$, 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.

References in corpus (4)

• Supports and extreme points in Lipschitz-free spaces
• Generalized transportation cost spaces
• On relations between transportation cost spaces and ${\ell }_1$
• Isometric copies of ${\ell }_{\infty }^n$ and ${\ell }_1^n$ in transportation cost spaces on finite metric spaces