TY - JOUR

T1 - Analysis on Laakso graphs with application to the structure of transportation cost spaces

AU - Dilworth, S. J.

AU - Kutzarova, Denka

AU - Ostrovskii, Mikhail I.

N1 - Funding Information:
The authors thank the referee for a very careful reading of the manuscript and for making numerous corrections and helpful suggestions which resulted in a much clearer presentation. The second author acknowledges the support from the Simons Foundation under Collaborative Grant No 636954. The third author gratefully acknowledges the support by the National Science Foundation Grant NSF DMS-1953773.
Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG part of Springer Nature.

PY - 2021/9

Y1 - 2021/9

N2 - 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.

AB - 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.

KW - Analysis on Laakso graphs

KW - Arens–Eells space

KW - Diamond graphs

KW - Earth mover distance

KW - Kantorovich–Rubinstein distance

KW - Laakso graphs

KW - Lipschitz-free space

KW - Transportation cost

KW - Wasserstein distance

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U2 - 10.1007/s11117-021-00821-w

DO - 10.1007/s11117-021-00821-w

M3 - Article

AN - SCOPUS:85112686806

VL - 25

SP - 1403

EP - 1435

JO - Positivity

JF - Positivity

SN - 1385-1292

IS - 4

ER -