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 arc also known as Arens-Eells, Lipschitzfree, 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 l(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 l(infinity)(n) have been found in terms of the canonical graph of X. The fact that TC spaces on diamond graphs do not contain l(infinity)(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.