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

TL;DR

This paper extends analysis of Laakso graphs by constructing orthogonal bases for cycle and cut spaces, estimating projection constants, and exploring the structure of associated transportation cost spaces, revealing their geometric properties.

Contribution

It introduces new orthogonal bases for Laakso graph cycle and cut spaces, and provides sharp estimates for projection constants and Banach-Mazur distances related to transportation cost spaces.

Findings

Projection constant estimates for Lipschitz spaces on Laakso graphs

Banach-Mazur distance lower bounds for transportation cost spaces

Examples of finite metric spaces containing  and  isometrically

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 $\operatorname{Lip}_0(\mathcal{L}_n)$, 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.