Isometric structure of transportation cost spaces on finite metric spaces

TL;DR

This paper investigates the isometric Banach space structure of transportation cost spaces on finite metric spaces, introducing new concepts and characterizations to understand their geometric properties.

Contribution

It introduces a new notion of roadmaps, simplifies representation proofs, and characterizes obstacles to embedding b spaces in transportation cost spaces.

Findings

Representation of TC spaces as quotients of b spaces on edges

Characterization of supports of maximal optimal roadmaps

Identification of obstacles preventing b embeddings

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.