Isometric structure of transportation cost spaces on finite metric spaces

Abstract

The paper is devoted to isometric Banach-space-theoretical structure of transportation cost (TC) spaces on finite metric spaces. The TC spaces are also known as Arens-Eells, Lipschitz-free, or Wasserstein spaces. A new notion of a roadmap pertinent to a transportation problem on a finite metric space has been introduced and used to simplify proofs for the results on representation of TC spaces as quotients of $$\ell _1$$ spaces on the edge set over the cycle space. A Tolstoi-type theorem for roadmaps is proved, and directed subgraphs of the canonical graphs, which are supports of maximal optimal roadmaps, are characterized. Possible obstacles for a TC space on a finite metric space X preventing them from containing subspaces isometric to $$\ell _\infty ^n$$ have been found in terms of the canonical graph of X. The fact that TC spaces on diamond graphs do not contain $$\ell _\infty ^4$$ isometrically has been derived. In addition, a short overview of known results on the isometric structure of TC spaces on finite metric spaces is presented.