This is an announcement for the paper “Generalized transportation cost spaces” by Sofiya Ostrovskahttps://arxiv.org/search/math?searchtype=author&query=Ostrovska%2C+S, Mikhail Ostrovskiihttps://arxiv.org/search/math?searchtype=author&query=Ostrovskii%2C+M.
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.
The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1902.10334