Generalized transportation cost spaces

Abstract

The paper is devoted to the geometry of transportation cost spaces and their generalizations introduced by Melleray, Petrov, and Vershik (2008). Transportation cost spaces are also known as Arens-Eells, Lipschitz-free, or Wasserstein $1$ spaces. In this work, the existence of metric spaces with the following properties is proved: (1) uniformly discrete infinite metric spaces transportation cost spaces on which do not contain isometric copies of ${\ell }_1$, this result answers a question raised by Cuth and Johanis (2017); (2) locally finite metric spaces which admit isometric embeddings only into Banach spaces containing isometric copies of ${\ell }_1$; (3) metric spaces for which the double-point norm is not a norm. In addition, it is proved that the double-point norm spaces corresponding to trees are close to ${\ell }_{\infty }^d$ of the corresponding dimension, and that for all finite metric spaces $M$, except a very special class, the infimum of all seminorms for which the embedding of $M$ into the corresponding seminormed space is isometric, is not a seminorm.

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Cited by in corpus (7)

• Supports in Lipschitz-free spaces and applications to extremal structure
• Lipschitz free spaces isomorphic to their infinite sums and geometric applications
• 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
• Tilings of the Hyperbolic Space and Lipschitz Functions
• Analysis on Laakso graphs with application to the structure of transportation cost spaces
• On the weak${}^{\ast }$ separability of the space of Lipschitz functions