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

Autor: Stephen J. Dilworth, Mikhail I. Ostrovskii, Denka Kutzarova
Rok vydání: 2021
Předmět:
Zdroj: Positivity. 25:1403-1435
ISSN: 1572-9281
1385-1292
DOI: 10.1007/s11117-021-00821-w
Popis: 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 $$\mathcal {L}_n$$ . 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 $${\text {Lip}}_0(\mathcal {L}_n)$$ , the space of Lipschitz functions on $$\mathcal {L}_n$$ . We deduce that the Banach–Mazur distance from $${\mathrm{TC}}\quad (\mathcal {L}_n)$$ , 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 $${\text {Lip}}_0(D_{n,k})$$ , 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.
Databáze: OpenAIRE
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