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

S. J. Dilworth, Denka Kutzarova, Mikhail I. Ostrovskii

Research output: Contribution to journalArticlepeer-review


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 Ln. 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 (Ln) , the space of Lipschitz functions on Ln. We deduce that the Banach–Mazur distance from TC(Ln), the transportation cost space of Ln, to ℓ1N of the same dimension is at least (3 n- 5) / 8 , which is the analogue of a result from [op. cit.] for the diamond graph Dn. We calculate the exact projection constants of Lip (Dn,k) , where Dn,k is the diamond graph of branching k. We also provide simple examples of finite metric spaces, transportation cost spaces on which contain ℓ∞3 and ℓ∞4 isometrically.

Original languageEnglish (US)
Pages (from-to)1403-1435
Number of pages33
Issue number4
StatePublished - Sep 2021


  • Analysis on Laakso graphs
  • Arens–Eells space
  • Diamond graphs
  • Earth mover distance
  • Kantorovich–Rubinstein distance
  • Laakso graphs
  • Lipschitz-free space
  • Transportation cost
  • Wasserstein distance

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Analysis
  • General Mathematics


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