Convexity of a Ball in the Gromov–Hausdorff Space / D. P. Klibus. // Vestnik Moskovskogo Universiteta. Seriya 1. Matematika. Mekhanika. 2018. № 6. P. 41-45
[Moscow Univ. Math. Bulletin. Vol. 72, N 2, 2017. P. 249-253].

The space \mathcal{M} of all nonempty compact metric spaces considered up to an isometry
and endowed with the Gromov–Hausdorff distance is studied. It is shown that each ball in \mathcal{M}
centered at a single-point space is convex in the weak sense, i.e., any two points of the ball
can be connected by a shortest curve lying inside the ball, but it is not convex in the strong sense,
i.e., not every shortest curve connecting some points of the ball lies inside the ball.
It is also shown that each ball of sufficiently small radius centered at a generic metric space
is convex in the weak sense.

Key words:
Gromov–Hausdorff metric, convex in the weak sense, convex in the strong sense.