Geodesic
Path connectedness of spheres in Gromov–Hausdorff space
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 2 (2019), pp. 42-46 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
@article{VMUMM_2019_2_a7,
     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},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMUMM_2019_2_a7/}
}
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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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[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