Complexity of distances between metric and Banach spaces

Marek Cúth, Michal Doucha, Ondřej Kurka

Published 2018-04-30Version 1

We investigate the complexity and reducibility between analytic pseudometrics coming from functional analysis and metric geometry, such as Gromov-Hausdorff, Kadets, and Banach-Mazur distances. This leads us to introduce the notion of Borel reducibility between pseudometrics which generalizes the standard Borel reducibility between definable equivalence relations and is a quantitative version of it, and orbit pseudometrics, the continuous version of orbit equivalences. Our results include the mutual bi-reducibility between Gromov-Hausdorff, Banach-Mazur, Kadets, Hausdorff-Lipschitz, net and Lipschitz distances, and their reducibility to the uniform distance. We show that $E_1$ is not reducible to equivalences given by these pseudometrics. Among our applications are the proofs that the distance-zero classes in these pseudometrics are Borel, extending the results of Ben Yaacov, Doucha, Nies, and Tsankov, and answering their question in negative whether balls in these distances are Borel. Besides that, we provide many other examples and problem areas to be looked at, which suggests that there is enough further possible development in this field.