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Branching geodesics of the Gromov--Hausdorff distance

In this paper, we first evaluate topological distributions of the sets of all doubling spaces, all uniformly disconnected spaces, and all uniformly perfect spaces in the space of all isometry classes of compact metric spaces equipped with the Gromov--Hausdorff distance. We then construct branching geodesics of the Gromov--Hausdorff distance continuously parameterized by the Hilbert cube, passing through or avoiding sets of all spaces satisfying some of the three properties shown above, and passing through the sets of all infinite-dimensional spaces and the set of all Cantor metric spaces. Our construction implies that for every pair of compact metric spaces, there exists a topological embedding of the Hilbert cube into the Gromov--Hausdorff space whose image contains the pair. From our results, we observe that the sets explained above are geodesic spaces and infinite-dimensional.