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Topological dimension of the Gromov-Hausdorff and Gromov-Prokhorov spaces
Hiroki Nakajima, Takamitsu Yamauchi, Nicolò Zava
The Gromov-Hausdorff distance is a dissimilarity metric capturing how far two spaces are from being isometric. The Gromov-Prokhorov distance is a similar notion for metric measure…
Extension of Gromov's Lipschitz order to with additive errors
Hiroki Nakajima
Gromov's Lipschitz order is an order relation on the set of metric measure spaces. One of the compactifications of the space of isomorphism classes of metric measure spaces equippe…
Principal bundle structure of the space of metric measure spaces
Daisuke Kazukawa, Hiroki Nakajima, Takashi Shioya
We study the topological structure of the space of isomorphism classes of metric measure spaces equipped with the box or concentration topologies. We consider the sca…
Topological aspects of the space of metric measure spaces
Daisuke Kazukawa, Hiroki Nakajima, Takashi Shioya
Gromov introduced two distance functions, the box distance and the observable distance, on the space of isomorphism classes of metric measure spaces and developed the convergence t…
A natural compactification of the Gromov-Hausdorff space
Hiroki Nakajima, Takashi Shioya
In this paper, we introduce a pseudometric on the family of isometry classes of (extended) metric spaces. Using it, we obtain a natural compactification of the Gromov-Hausdorff spa…
Convergence of group actions in metric measure geometry
Hiroki Nakajima, Takashi Shioya
We generalize the box and observable distances to those between metric measure spaces with group actions, and prove some fundamental properties. As an application, we obtain an exa…