Generalized Transportation Cost Spaces

The 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 Cúth 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 _∞ ^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.