Geodesic
Convexity of a ball in the Gromov–Hausdorff space
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 6 (2018), pp. 41-45 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
@article{VMUMM_2018_6_a5,
     author = {D. P. Klibus},
     title = {Convexity of a ball in the {Gromov{\textendash}Hausdorff} space},
     journal = {Vestnik Moskovskogo universiteta. Matematika, mehanika},
     pages = {41--45},
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     url = {http://geodesic.mathdoc.fr/item/VMUMM_2018_6_a5/}
}
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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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