Ostrovska, SofiyaOstrovskii, Mikhail I.Mathematics2024-07-052024-07-05201991660-54461660-545410.1007/s00009-019-1433-82-s2.0-85075802841https://doi.org/10.1007/s00009-019-1433-8https://hdl.handle.net/20.500.14411/3463The paper is devoted to the geometry of transportation cost spaces and their generalizations introduced by Melleray et al. (Fundam Math 199(2):177-194, 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 metric spaces such that transportation cost spaces on them do not contain isometric copies of l(1), this result answers a question raised by Cuth and Johanis (Proc Am Math Soc 145(8):3409-3421, 2017); (2) locally finite metric spaces which admit isometric embeddings only into Banach spaces containing isometric copies of l(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 l(infinity)(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.eninfo:eu-repo/semantics/openAccessArens-Eells spaceBanach spacedistortion of a bilipschitz embeddingEarth mover distanceKantorovich-Rubinstein distanceLipschitz-free spacelocally finite metric spacetransportation costWasserstein distanceArticleQ2Q2166WOS:000508589800002