Geodesic
Convexity of a ball in the Gromov–Hausdorff space
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 6 (2018), pp. 41-45 
D. P. Klibus. Convexity of a ball in the Gromov–Hausdorff space. Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 6 (2018), pp. 41-45. http://geodesic.mathdoc.fr/item/VMUMM_2018_6_a5/
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     author = {D. P. Klibus},
     title = {Convexity of a ball in the {Gromov{\textendash}Hausdorff} space},
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     url = {http://geodesic.mathdoc.fr/item/VMUMM_2018_6_a5/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper we study the space $\mathcal{M}$ of all nonempty compact metric spaces considered up to isometry equipped with the Gromov–Hausdorff distance. We show that each ball in $\mathcal{M}$ with the center at the one-point space is convex in the weak sense, i.e., any two points of such a ball can be joined by a shortest curve that belongs to this ball, and is not convex in the strong sense: it is not true that every shortest curve joining the points of the ball belongs to this ball. It is also shown that a ball of sufficiently small radius with the center at a space of general position is convex in the weak sense.

[1] Edwards D., The Structure of Superspace, Studies in Topology, Academic Press, N.Y.–S.F.–L., 1975 | MR

[2] Gromov M., “Groups of polynomial growth and expanding maps”, Publ. math., 53 (1981), 53–73 | DOI | MR

[3] Burago D. Yu., Burago Yu. D., Ivanov S. V., Kurs metricheskoi geometrii, Institut kompyuternykh issledovanii, M.–Izhevsk, 2004 | MR

[4] 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

[5] Ivanov A. O., Tuzhilin A. A., Geometriya rasstoyanii Khausdorfa i Gromova–Khausdorfa: sluchai kompaktov, Izd-vo Popechitelskogo soveta mekh.-mat. f-ta MGU, M., 2017

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

[7] Ivanov A. O., Tuzhilin A. A., Local Structure of Gromov–Hausdorff space near finite metric spaces in general position, 2016, arXiv: 1611.04484 | MR