Convexity of a ball in the Gromov--Hausdorff space
Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 6 (2018), pp. 41-45
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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--Hausdorff} space},
journal = {Vestnik Moskovskogo universiteta. Matematika, mehanika},
pages = {41--45},
publisher = {mathdoc},
number = {6},
year = {2018},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/VMUMM_2018_6_a5/}
}
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/