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arXiv · 1910.03625

On relations between transportation cost spaces and $\ell_1$

Sofiya Ostrovska·Mikhail I. Ostrovskii
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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.

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Sofiya Ostrovska, Mikhail I. Ostrovskii. 2020-06-22. On relations between transportation cost spaces and $\ell_1$. https://doi.org/10.1016/j.jmaa.2020.124338

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