Geodesic
Noncompactness of segments in the Gromov--Hausdorff space
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 5 (2021), pp. 3-8

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We study properties of segments in the Gromov–Hausdorff metric space. A segment is a subset of a metric space consisting of points lying between two given points. We prove that any segment in the Gromov–Hausdorff space with endpoints being non-isometric compact metric spaces contains an element that is a compact metric space with at least one isolated point. Using this theorem and Gromov's precompactness criterion, we prove that any nondegenerate segment in the Gromov–Hausdorff space is not a compact set. 
@article{VMUMM_2021_5_a0,
     author = {O. B. Borisova},
     title = {Noncompactness of segments in the {Gromov--Hausdorff} space},
     journal = {Vestnik Moskovskogo universiteta. Matematika, mehanika},
     pages = {3--8},
     publisher = {mathdoc},
     number = {5},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VMUMM_2021_5_a0/}
}
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O. B. Borisova. Noncompactness of segments in the Gromov--Hausdorff space. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 5 (2021), pp. 3-8. http://geodesic.mathdoc.fr/item/VMUMM_2021_5_a0/