On relations between transportation cost spaces and ${\ell }_1$

Abstract

The present paper deals with some structural properties of transportation cost spaces, also known as Arens-Eells spaces, Lipschitz-free spaces and Wasserstein spaces. The main results of this work are: (1) A necessary and sufficient condition on an infinite metric space $M$, under which the transportation cost space on $M$ contains an isometric copy of ${\ell }_1$. The obtained condition is applied to answer the open questions asked by Cúth and Johanis (2017) concerning several specific metric spaces. (2) The description of the transportation cost space of a weighted finite graph $G$ as the quotient ${\ell }_1(E(G))/Z(G)$, where $E(G)$ is the edge set and $Z(G)$ is the cycle space of $G$. This is a generalization of the previously known result to the case of any finite metric space.

References in corpus (8)

• On the structure of Lipschitz-free spaces
• Lipschitz-free spaces over ultrametric spaces
• Generalized transportation cost spaces
• Embeddings of Lipschitz-free spaces into ${\ell }_1$
• Isometric representation of Lipschitz-free spaces over convex domains in finite-dimensional spaces
• Isometric embedding of ${\ell }_1$ into Lipschitz-free spaces and ${\ell }_{\infty }$ into their duals
• Isometric copies of ${\ell }_{\infty }^n$ and ${\ell }_1^n$ in transportation cost spaces on finite metric spaces
• Some natural subspaces and quotient spaces of $L^1$

Cited by in corpus (3)

• Points of differentiability of the norm in Lipschitz-free spaces
• Lipschitz algebras and Lipschitz-free spaces over unbounded metric spaces
• Analysis on Laakso graphs with application to the structure of transportation cost spaces