Geodesic
Path connectedness of spheres in Gromov–Hausdorff space
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 2 (2019), pp. 42-46 
R. A. Tsvetnikov. Path connectedness of spheres in Gromov–Hausdorff space. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 2 (2019), pp. 42-46. http://geodesic.mathdoc.fr/item/VMUMM_2019_2_a7/
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     author = {R. A. Tsvetnikov},
     title = {Path connectedness of spheres in {Gromov{\textendash}Hausdorff} space},
     journal = {Vestnik Moskovskogo universiteta. Matematika, mehanika},
     pages = {42--46},
     year = {2019},
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     url = {http://geodesic.mathdoc.fr/item/VMUMM_2019_2_a7/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

The path connectedness of spheres in Gromov–Hausdorff space is studied. The following two assertions are proved: (1) each sphere centered at one-point space is path connected; (2) for any metric space $X$ there exists a number $R_X$ such that each sphere with the center at $X$ and radius greater than $R_X$ is path connected.

[1] Burago D. Yu., Burago Yu. D., Ivanov S. V., Kurs metricheskoi geometrii, M., 2004

[2] Ivanov A. O., Iliadis S., Tuzhilin A. A., Realizations of Gromov–Hausdorff distance, 2016, arXiv: 1603.08850 | MR

[3] Ivanov A. O., Nikolaeva N. K., Tuzhilin A. A., The Gromov–Hausdorff metric on the space of compact metric spaces is strictly intrinsic, 2015, arXiv: 1504.03830 | MR