# Transportation cost spaces on finite metric spaces

• Denka Kutzarova (University of Illinois at Urbana-Champaign; Bulgarian Academy of Sciences)
• Monday 17 June 2019, 11:10-12:00
• Seminar Room 1, Newton Institute.

ASCW03 - Approximation, sampling, and compression in high dimensional problems

Transportation cost spaces are studied by several groups of researchers, for different reasons and under different names. The term Lipschitz-free spaces is commonly used in Banach space theory.
We prove that the transportation cost space on any finite metric space contains a large well-complemented subspace which is close to $\ell_1n$.
We show that transportation cost spaces on large classes of recursively defined sequences of graphs are not uniformly isomorphic to $\ell_1 n$ of the corresponding dimensions. These classes contain well-known families of diamond graphs and Laakso graphs.
In the particular case of diamond graphs we prove that their cycle space is spanned by even levels of Haar functions. It is curious that the subspaces generated by all the even/odd levels of the Haar functions also appear in the study of quasi-greedy basic sequences in $L_1[0,1]$.
This research is joint with Stephen Dilworth and Mikhail Ostrovskii.

This talk is part of the Isaac Newton Institute Seminar Series series.