Convexity of Balls in Gromov--Hausdorff Space

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 center at the one-point space is convex in the weak sense, i.e., every two points of such a ball can be joined by a shortest curve that belongs to this ball, however, such a ball 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. We also show that a ball of sufficiently small radius with center at a space of general position is convex in the weak sense.